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Coverage of the phrase in recent reference works

J. E. Lighter, The Random House Historical Dictionary of American Slang (1994) takes a cautious approach to the question of how "get [one's] goat" originated:

get (someone's) goat despite several attempted explanations, the inspiration behind this phrase remains unknown 1. to anger or annoy (someone).

[First four relevant citations:] 1904 Life in Sing Sing 248: Goat. Anger. 1908 in Fleming Unforget[table] Season 184: The supreme contempt shown...evidently got the "goat" of Mr. Frederick Clarke. 1908 in H.C. Fisher A. Mutt 46: To have one's goat is to have one buffaloed or the Indian sign on one's contemporaries. 1908 Atlantic (Aug.) 223: A little detraction will "get their goat."

Christine Ammer, The American Heritage Dictionary of Idioms (1997) is likewise cautious about the term's origin:

get someone's goat Annoy or anger someone, [example omitted]. The origin of this expression is disputed. H. L. Mencken held it came from using a goat as a calming influence in a racehorse's stall and removing it just before the race, thereby making the horse nervous. However, there is no firm evidence for this origin. c. 1900

Chrysti Smith, Verbivore's Feast: Second Course indicates that H.L. Mencken advanced the "somewhat fanciful theory" of a confidence-boosting relationship between racehorses and goats in one of his books (presumably an edition of The American Language) in 1945.

Nigel Rees, Cassell's Dictionary of Word and Phrase Origins (1996) is skeptical of this theory, too:

get one's goat, to To be annoyed by something. 'The way she carries on, that really gets my goat!' Apparently another Americanism that has passed into general use (and current by 1910), this expression can also be found in French as prendre la chèvre, 'to take the milch-goat'. One is always suspicious of explanations that go on to explain that, of course, goats were very important to poor people and if anyone were to get a man's goat ... etc. One is even more unimpressed by the explanation given by the Morris Dictionary of Word and Phrase Origins (1977): 'It used to be a fairly common practice to stable a goat with a thoroughbred [horse], the theory being that the goat's presence would help the high-strung nag to keep its composure. If the got were stolen the night before a big race, the horse might be expected to lose its poise and blow the race.'

Robert L. Shook in The Book of Why (1983) wonders, interestingly, whether it has anything to do with a 'goatee' (a beard like a goat's). If you get someone by the goat, it would certainly annoy them.

All one can do is point to the number of idioms pointing to goats—'act the goat', 'giddy goat', 'scapegoat', and, once more, emphasize the alliteration. Another version is 'to get one's nanny-goat'.

Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins (1997), however, is less dismissive of the comforting goat theory:

It's as good an explanation as any, but isn't supported by much evidence. Jack London was the first to record the expression, in his novel R (1912), tough the usage there has nothing to do with racing. Attempts to connect the goat in the phrase with the scapegoat of Hebrew tradition; with the word goad, "to anger, irritate"; and to an old French phrase prendre la chèvre, literally meaning "to take the goat," which dates back to the 16th century and certainly took a long time making the journey to America if it is the source of our expression.

It seems relevant to note that the exhaustive Farmer & Henley, Slang & Its Analogues, volume 3 (1893) has no slang entry for goat that even remotely suggests the meaning "to anger or annoy." This suggests that the idiom "get one's goat" arose fairly close to the time (1907–1908) when it burst onto the national scene in its still-current sense.

Occurrences of the idiom in the context of sports

A Google Books search finds a number of matches from as early as 1907, including some that Lighter (above) cites. Since so many involve a sporting context, I will divide my coverage by sport.

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Baseball

Two of the earliest three citations in Lighter for the phrase "get [one's] goat" come from articles or books about baseball, so that may be a sensible place to begin.

From William Kirk, account of a baseball game between the Pittsburgh Pirates and the New York Giants, New York American (August 26, 1908), reprinted in Gordon Fleming, The Unforgettable Season (1981):

The Pirates do not look so terrible now. One week ago it was the general impression that if the Giants were to win the pennant they would have to beat Pittsburgh. But [John] McGraw, crafty little gent that he is, gave out an interview and said: "I do not fear Pittsburgh in the least. It is Chicago we must beat to win the flag." This surprising statement, coming just when the Pirates were soaring and the Cubs losing, astounded and unnerved the Pittsburghers. The supreme contempt shown by Manager Mac for the club on top evidently got the "goat" of Mr. Frederick Clarke [the Pirates' manager]. McGraw does not know a thing. He goes to the races too much.

From Rollin Hartt, "The National Game," in Atlantic Monthly (August 1908):

For artists they [baseball players] are—sensitive as violinists, "temperamental" as painters, emotional as divas. A little detraction will "get their goat," a little adulation prepare them to walk upon pink clouds. As the Presbyterian said of the Methodists, they are "up attic or down cellar all the while."

A slightly later instance, Roe Fulkerson, "The Old Groceryman" in The [San Francisco] Retail Grocers Advocate (July 12, 1912) credits baseball as the source of the idiom, which, if nothing else, illustrates how widely used "get [someone's] goat" was with baseball at that time:

"The great national pastime of Ty Cobb has brought into the language of this land a few expressions which would not particularly adorn a thesis to be read before a society of blue stockings, but which fit mighty well in business circles.

"One of these expressions relates to a certain goat.

"When a baseball pitcher, time after time, beats a certain team, that team gets into a state of mind where the mere appearance of that pitcher in the box has them beaten to a frazzel before the game starts. It is then said that this particular pitcher 'has that team's goat.' That's just what's the matter with you, Bill, your competitors have got your goat.

The phrase is used in its modern sense in all three of these sources—and other baseball-related instances occur in Letters from a Baseball Fan to His Son (1910), The Rival Pitchers (1910), "A Bush League Hero" (1912),Those Smith Boys on the Diamond (1912), Baseball Joe on the School Nine (1912), Baseball Joe of the Silver Stars (1912), Baseball Joe at Yale (1913), and Lefty o' the Blue Stockings (1914)—but there is no evident link to an actual goat in any of them.

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Boxing

The use of "got his goat" in the sense of "made him lose his composure" appears in Richard Barry, "The Prize Ring" in Pearson's Magazine (July 1910):

What Nelson had been working for twenty rounds to accomplish cam to pass instantly [with an unexpected taunting remark]. Hyland's goat passed from his stable and Fighting Dick threw himself on Nelson's spear. ...

For years, Nelson did these things in the ring, but he tried it once too often ten months later with Ad Wolgast, who returned taunt for taunt and who finished by wrenching Nelson's title from him. At that Nelson never lost his goat and was never knocked down. He merely failed to get the other fellow's goat.

That same article asserts that the term came from horse racing:

Freddie Welsh, the present lightweight champion of England, a vegetarian with puny hips, watery eyes and a weak mouth, who has never been knocked out, puts his advice to aspirants in three words, "Get his goat!"

Originally this phrase was racing slang. To keep a racehorse from going stale a trainer frequently quarters with him a goat, for the pet relieves the thoroughbred of his loneliness. But intriguers have found that by stealing a goat from a horse a day or two before a great race he can be thrown out of condition. The loss of his favorite companion annoys the horse and he goes into the big event in a highly feminized state of nerves. So, to "get his goat" is to remove his confidence.

To similar effect is this explanation from E.B. Osborn, "The Revival of Boxing in the Nineteenth Century," in The Twentieth Century (October 2011):

One of the chief preliminary points is to 'get the goat' of his opponent; by interpretation, so to jar his feelings that his mind will not be altogether in his work when the gong sounds for hostilities to begin. A disgusting taunt, a contemptuous smile, a refusal to shake hands, a preposterous bet, an elaborate wasting of time—these re some of the tricks employed by the would-be goat-getter.

Oscar Battling Nelson uses the phrase himself in Life, Battles and Career of Battling Nelson, Lightweight Champion of the World (1908):

Britt is a strong, game, clever fighter. The only man that ever made Britt show the white feather was Joe Gans. I gave him a much worse beating than Gans did, but the minute he saw the black fellow in the ring he practically threw up his hands and admitted defeat. In his fight with all the other lightweights Britt was game to the core. I never could exactly understand why he let Gans get his goat.

...

As will well be remembered by the many patrons who attended this club [the National Athletic Club] on the memorable night of March 14, 1906, I kept Terry McGovern waiting in the cold on the raised platform for about three quarters of an hour. Knowing that McGovern was very, very nervous and easily "riled," I took my time in putting the tape on my hands purposely to get his goat. I certainly succeeded.

...

We had hardly fought thirty seconds when I could tell that I had McGovern's goat. He was nervous and held on. He was afraid to fight his usual fight of rushing from start to finish.

But an even earlier instance of "got his goat" appears in Rex Beach, "The Fight at Tonopah," in Everybody's Magazine (April 1907):

In a quiet interval between rounds I heard a reporter dictating high-class pugilistic literature:

"Herman's work in the fifth was classy and he fought all over the place. He stabbed the Dinge in the food-hopper three times and all but got his goat, then missed a right swing to the butler's pantry by an inch. If he had coupled, it would have been the sunset glow for Dahomey, but Gans didn't fall for the gag, not hardly. He ripped an upper through the Yiddish lad and put him on the hop with a right cross."

As is the case with baseball, the phrase "get someone's goat" seems endemic to boxing by 1910, though the sport has no direct connection to an actual goat. I find it significant that a well-informed journalist writing about boxing in 1910 is under the impression that the expression originated with horse racing. After all, the earliest Google Books match I could find for the phrase was only three years earlier, so it isn't as though people were speculating on the origin of a phrase that had been in common parlance for more than a century (as is the case today).

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Horse racing

An early instance of the "goat as calming influence on a racehorse" explanation appears in Elbert Hubbard, "Got His Goat," in The Fra: A Journal of Affirmation (1912):

"Got His Goat"

There are lots of highly educated people with college degrees who do not know the origin of the classic expression, “Got his Goat.” This fine literary phrase, so far-reaching in its psychic import, originated in East Aurora[, Illinois.] It was first used by Ali Baba.

...

In the stable of Ed Geers was a very fast horse known as Prince Regent. This horse was of a very nervous disposition, and when in a box stall by himself—especially in a strange place—fretted and suffered from homesickness[.] Ed Geers bought a goat from Ali Baba and put it in the stall with the horse, and the horse and the goat became very chummy.

...

There came a day when this horse was to trot for the Futurity Stakes in Buffalo. All went well, and it looked as if Prince Regent had a sure thing. The bets were all in favor of Prince Regent.

But at midnight on the day before the race, some one got in the barn and stole the goat, and the horse so pined for his mate that he seemed to lose heart, and by the afternoon when the time came for the horses to be called, Prince Regent lost his nerve.

"What's the matter with that horse?" somebody said, as thy noticed the animal's drooping ears.

Why somebody has got his goat," said Ali Baba.

And so the phrase has gone clattering down the centuries, like a tin kettle to a dog's tail. It has been fixed in the current coin of speech.

"We have got his goat." That means the man is done for and out of the game.

The name Ali Baba lends an air of fantastical improbability to this story, and the idea that an internationally popular expression came into being in such a picturesque way among longtime friends of the reporting author should set off alarm bells in the head of any skeptical reader. Nevertheless, the story has a number of data points in reality: For example, Ed Geers really did own a stable of racehorses, and Prince Regent was a notable race horse (a trotter, actually, as many racing horses were in those days) of the 1890s, according to Badger, "The Horses at the Columbian," in Wallace's Monthly (April 1893). An item in the Wichita [Kansas] Daily Eagle (May 27, 1890) notes:

The princely sum of $40,000 has been refused for Prince Regent, a horse which in 1889 proved himself a race horse of the first class, and got a 4-year-old mark of 2:21¼. Prince Regent is a son of Mambrino King, called the handsomest horse in the world; ...

And the St. Paul Daily Globe (April 13, 1891) reported the horse's death with the headline and and subhead "Prince Regent Dead: The Great Stallion Passes Away Near Buffalo."

According to the same Elbert Hubbard, writing on other occasions, Ali Baba was a respected a local stable hand. From Elbert Hubbard, "Heart to Heart Talks with Philistines by the Pastor of His Flock," in The Philistine (November 1907):

East Aurora has produced three great men[.] These are Cicero J. Hamlin, Ed. Geers, and Charles A. Cyphers.

In saying these things I fully realize that I am laying myself open to the acerbity of Ali Baba, Deacon Buffum, and Uncle Billy Bushnell, all of whom, even if not great men, are surely industrious persons.

So the jury is out on the whole "stolen goat stablemate" theory, and much more so on the friends-of-Elbert-Hubbard reminiscence. But again, the nearness of the story to the earliest recorded instances of the idiom is a point in its favor. And the fact that a reporter was stating as fact in 1910—two years before Hubbard told his tale—that goats were then sometimes kept as companions for race horses and that people betting against such horses did sometimes steal the goat before a race to upset the horse at least shows that a connection between "get [one's] goat" and the story about kidnapped goat companions existed from very early in the life of "get [one's] goat" as a popular idiom. These are not trivial points in the story's favor.

Early occurrences of the idiom in nonsporting contexts

One of the earliest instances of "get one's goat" in a nonsporting context that my Google Books searches turned up is from Burke Jenkins, "The Anti-Climax of a Bad Man," in The Argosy (September 1907):

"'Don't shoot, Hankly,' it [a voice from the shubbery] said; 'I'll come peaceable,' Then, sir, out crawled Kinston, scratched with briers and clean out of heart. I played the card which had fallen right in my hand—bound him up secure, and brought him along. Yer see, I'd kind of 'got his goat' by walking up to that there bloody gun of his."

From L.J.W., "The Hurry-Up Men," in The National Engineer (February 1908):

The slang phrase "got his goat" is applicable to some phases of stationary engineering. The coiner of the phrase means the goat to signify temper which rises over some incident or happening not to the liking of the goatee (not copyrighted).

If the writer possesses a goat it is mostly due to the habits and actions of certain class of engineers to be described. Every neighborhood boasts of at least one of the type. We will call him Mr. Hurryup.

From John McIntyre, "For the Glory of Monsignor," in The Reader (February 1908):

"Father Augustine puts it very succinctly : A bishop is not only required to be a devout and learned man. He must have the ability to do things, the magnetism to compel others. This matter is a good test of monsignor's capabilities in that direction. Let him fail to succeed upon this mission, as we might call it, and he'll never be a bishop.

...

"Do you believe it?" inquired Hopkins, after a space.

“Sure,” replied Riley. “It sounds just like the truth. If he don't raise the money, they'll get his goat.”

This is the only mention of goat in the story, so it is clearly being used metaphorically, but the context has nothing in common with a situation where someone loses composure. Evidently it means something quite different to the two boys who use it in this story than it means to Battling Nelson in his autobiography.

One other interesting early use of the phrase occurs in "Chicago, Ill." union notes in The Typographical Journal (November 1909):

At the last meeting of No. 16 ex-President Colbert proceeded to get the goat of President Knott. While waiting for the representative of the hatters' union, who had been granted the privilege of addressing the meeting, Mr. Colbert asked for the privilege of the floor, and in a neat little speech presented to President Knott a gavel the gift of many friends in No. 16. The gavel is made of ebony, inlaid with pearl, and is a magnificent work of art, every bit of work on it being done by union men and bears the label. Suitable inscriptions are on the face of the gavel. Mr. Knott was taken completely by surprise, and it was some moments before he could get his "goat" under control and thank the members of No. 16 for such a magnificent gift.

Here, goat is evidently being used synonymously with "emotions"; a broader reading could interpret it as meaning "equanimity," I suppose, but the rest of the sentence's wording doesn't quite work with that reading.

Conclusions

The origin of the idiom "get [someone's] goat" remains uncertain; but of the competing claimants, horse racing seems to me to have the strongest case—and the "stolen goat companion" theory is supported by earlier and more-varied evidence than some major references from the pre-Internet age were aware of. In particular, a boxing story from 1910 asserts that some stables did house goats with high-strung horses even at that time—and claims that miscreants seeking to skew the results of an impending race did sometimes steal a goat from its shared quarters with a race horse. This report is especially striking in that the earliest Google Books matches for various forms of "get [one's] goat" date back to 1907—only three years earlier. So it appears that, accurate or not, the stolen goat explanation has been tied to the phrase since before most people were even aware of it as an idiom.

The other striking thing that the Google Books search results underscore is how rapidly the expression went from being virtually unknown in U.S. publications to becoming a standard expression in three extremely popular sports of the era: baseball, horse racing, and boxing.

According to The Phrase Finder:

The phrase originated in the US and the first entry in print that [...]
comes from a fanciful story about a burst water pipe that was printed
in the US newspaper The Stevens Point Daily Journal, May 1909:

« Wouldn't that get your goat? We'd been transferring the same water
all night from the tub to the bowl and back again. »

[...]

A commonly repeated story which purports to explain the phrase's
origin is that goats were placed with racehorses to keep them calm.
When ne'er-do-wells who wanted the horse to race badly removed it,
that is, they 'got someone's goat', the horse became unsettled and ran
badly. That's just the sort of tale that gets the folk etymology
juices running. Let's just say that there's no evidence to support
that story.

I think "getting my goat" likely originated with someone yanking on someone's goatee, which would be very annoying.

Wikipedia:
Until the late 20th century, the term goatee was used to refer solely to a beard formed by a tuft of hair on the chin—as on the chin of a goat, hence the term 'goatee'.[1]

Etymonline says the saying is American English from 1910. As it is a very twisted saying that makes no sense I assume the saying developped in spoken language long before a first written incidence was recorded.

Goat, the animal, makes no sense; so it is possible it was a totally different word that changed its shape and finally was replaced by a common and known word. At first I had no idea at all what might be behind goat and would go in the direction of annoyance. The only thing that came to mind after some time was German Da kommt mir der Magen hoch - It uplifts my stomach.

Perhaps the Greek word for stomach, gastaer (ae is eta) that we have in the medical term gastritis, has developed a short form (in spoken language) as *gast/*gost which became goat. This is a mere idea of mine, so please don't ask for references, and as etymonline has only an assumption about the origin marked with "perhaps" I don't discredit etymonline as someone said.
On the contrary, I have etymonline in high esteem, it is a tremendous work of reference.

But perhaps someone has another and better idea about the origin of goat.
http://www.etymonline.com/index.php?allowed_in_frame=0&search=goat&searchmode=none

I forgot to say the idea of the saying might have been: It gets my stomach
(in turmoil).

Edit: This is a forum about horses. Especially about the question keeping horses and goats together. See for yourself.
http://www.horseforum.com/farm-forum/keeping-horses-goats-36455/

Maybe our understanding of what the term actually meant at one time could be off the mark. I know some about farming that might ad to the discussion.

Goats are wont to give you a charging attack for little to no reason.

If you've lived around goats, as poor people frequently did in the past. You know that you only have to be standing there minding your own business, maybe having a conversation with others, when a goat is enticed to butt you. It happens often enough so that anyone with goats would instantly recognize the analogy.

Maybe the idiom passed from farmers to sports most likely. When someone said "That gets my goat" it means that someone or something is inviting my attack. But as society passed from more agrarian to industrialized fewer and fewer people knew that goats are ornery creatures, so the etymology was generally lost. Or twisted to a slightly different meaning.

The deeper understanding of that is maybe, you never know what will set a goat off and cause it you butt you. Could be any number of triggers. My Granparents said it frequently enough to mean that and they were salt of the earth idiom kind of people. I know that's anecdotal but there it is.

So from the top of the page:
" The phrase originated in the US and the first entry in print that [...] comes from a fanciful story about a burst water pipe that was printed in the US newspaper The Stevens Point Daily Journal, May 1909:

"Wouldn't that get your goat? We'd been transferring the same water all night from the tub to the bowl and back again."

As the analogy would go. If we all had a goat within us, some situations would cause that goats main characteristic to come out. That of a butting attack. So in the above, seeing the futility of that pipe situation would entice ones inner goat to butt it out of frustration.

That's how metaphors work, yes? If we are using some animal to define some characteristic within us, the animal is used for it's predominate characteristic that we all immediately recognize and are familiar with.

Believe me the one characteristic, above all others, that all farmers associate with goats is the horns and that darned insistence on butting you now and then.

I'm not well educated. So maybe someone else can articulate this a little better. But the idiom "gets my goat" has always, to my family, meant that something has triggered you into responding in a somewhat aggressive way. It may have been something that angered or annoyed you. But sometimes for no particular reason.

I always thought that the term we use today "being triggered" was similar though not exactly the same thing. Of course this could all be the inarticulate ramblings of a country yokel too. Just throwing that out there.

'Got somebody's goat': Of two early uses in 1900, 16 Nov and 26 Nov (both paywalled), the 16 Nov use comes complete with an origin story. Presumably based on the anonymous reporter's personal and professional experiences, the origin story's early appearance has that advantage, as well as the advantage of being first-hand, over more recent speculation.

"HE HAS HIS GOAT."

The Latest New York Phrase Applied

to Certain Friendly Conditions.

 Sundry phrases at various times have been used to describe the influence which, in a friendly and unobjectionable way, one man is able to exercise over the doings of another. For many years the local New York expression in such a case was "He has the loan of him." Accordingly, when one man had "the loan of" another he was able to guide his course, says the New York Sun.

 It is the fate of New York colloquialisms that, though abruptly taken up, they are as summarily and capriciously abandoned, and the popularity of "the loan of" was not so great as to prevent its being superseded by another phrase of like import, in very general use until recently. This phrase was "Put me next." One reason of its popularity was the fact that it was subject to some variation. Thus an applicant for the favor or support of another asked to be "put next." A person who was already in the enjoyment of such good will was described as "being next," and a more chipper form of the expression was "he's next."

 The latest phrase used to convey this meaning is "He's got his goat." The meaning of the phrase is that the subject of the comment exercises large influence with another — he has the loan of him; he's next. A reason for this phrase is to be found, probably, in the summary disappearance from visible observation in New York of the goat of the rocks, a familiar figure in upper New York before the pressure of population obliterated the last traces of shantytown. In former days one of the most cherished possessions of squatters was a goat. Goats in New York at that period were of very little account, but since the city has grown to the extent of exclusion improved lots, the goat has become somewhat of a rarity. Harlem, therefore, appears to be the place of origin of the phrase, the immediate popularity of which is no doubt to some extent enhanced by the use of the word goat in Masonic circles in connection with a phase of the conditions of initiation which is diverting to every one but the initiated member. Whether this will furnish any reason for a longer continuance in popularity of "he's got his goat" than its predecessors have enjoyed remains to be seen.

The meaning of 'got somebody's goat', as detailed by the reporter, may be summarized as "got influence over somebody", a meaning which is compatible with the later but not invariable semantic development of "got influence over somebody, by angering, irritating, annoying or exasperating".

Having defined the special sense of the phrase, the reporter goes on to speculate that the "probable reason" for the development of that sense is the disappearance from Harlem of formerly common goats, along with the importance of those goats to impoverished squatters.

A garbled version of the early reporter's origin story is mentioned by Michael Quinion at World Wide Words (1999-2014), only to be discarded out of hand, possibly because the evidence from 1900 was not in Quinion's hand at the time:

It has been claimed that at one time some residents of Harlem in New York kept goats and thereby annoyed their neighbours, an explanation that fails to satisfy.

I find the vanished goats of Harlem — a lost source of food (milk, cheese and, ultimately, meat), clothing (wool), (body) heat, and companionship for destitute squatters — more satisfying as the origin of a phrase denoting the influence gained by one person as a result of another's loss of a goat, but the 26 Nov 1900 use could be construed as support for the theory that goats as naval (and other maritime) shipboard mascots sponsored the phrase.

The 26 Nov 1900 use of 'got somebody's goat' is reprinted from a letter dated 09 Nov, mailed from Gibralter by a navy sailor, Roy Krebs, on board The Kentucky bound for Hong Kong out of New York as part of the US response to the Boxer Rebellion (1899-1901).

"My enlistment expires June 15, 1904," writes Krebs, "when I will return to Atchison [Kansas] unless some Boxer gets my goat."

The meaning of "gets my goat" in Kreb's use seems likely to be "kills or seriously wounds me". That meaning is a far cry from the meaning noted by the reporter in the 16 Nov article, whereby the "friendly and unobjectionable" influence exercised by one man "over the doings of another" is conveyed.

I was able to corroborate the reporter's account of the history of goats in Harlem. For example, a headline on p. 9 of the New-York Tribune (New York, New York) 01 Sep 1889 (paywalled) mentions the loss:

UP IN BUSY HARLEM.

RAPID GROWTH OF THE PLACE,

GOATS AND ROCKS REPLACED BY HANDSOME HOMES.

 ...It was customary to refer to Harlem as an out-of-the-way place, the principle features of which were rocks and goats. ... If a Harlemite were to be asked where the laborers and the goats had gone, he would probably say that as the American Indian had been forced to the westward before the march of civilization, so had the laborer of Manhattan Island and his goats departed....

The disappearance of the cherished goats and their impoverished owners is also mentioned in 1897 (paywalled), on p. 3 of The Butte Daily Post (Butte, Montana), 14 May:

 To the observant citizen of New York returning to his native city after a sojourn of many years in other lands nothing is more striking than the mysterious absence of the belligerent and thrifty goats from the rock bound ranges of the Harlem and its environs. The goat and the squatter have vanished together from...where they once held undisputed control, and the traveler coming back...marvels at their disappearance, vaguely wondering whether it signifies migration or extinction.

As late as 1903 (paywalled), on p. 8 of The Sun (New York, New York), 19 Apr, the loss was still deplored: "Harlem has lost its goats, its rock-perched shanties, ...".

Without more context, a seemingly related definition of 'goat' supplied by Number 1500 in the 1904 book Life in Sing Sing (Sing-Sing is a New York prison) cannot be interpreted confidently:

Goat. Anger; to exasperate.

The close relationship of that use of the slang word 'goat' in Sing-Sing prison with the (perhaps later) semantic development of 'got somebody's goat' into "angered, annoyed, irritated, or exasperated somebody" is suggestive. However, as Number 1500 puts it, "the only apparent test of admission [to slang vocabulary is that] the word in sound must have some relation with the sense it is intended to convey" (p. 245). Thus, the meaning of 'goat' in Sing-Sing slang may rely only on the consonance of 'goat' and 'goad', and may have no bearing whatsoever on the meaning of 'got somebody's goat'.

Three instances of the complete phrase, 'to get [somebody's] goat', crop up in print in 1905. The effectively simultaneous print dates of the instances, 14 Oct, 28 Oct, and 18 Nov, neither suggest nor contradict any conjectures about the origin of the phrase, especially as they are five years distant from the uses in 1900.

The three 1905 uses, two of which are associated with US navy sailors, do offer some slender support for the theory that the 'got [somebody's] goat' phrase originated in the US navy with reference to goat mascots. Similarly, the association with boxing offers slender support for the theory that the phrase originated in boxing slang. About this latter theory, more later, but the boxing and the naval origin stories are undermined by the effectively simultaneous uses in both 1900 and 1905 with more general contexts.

The first, 14 Oct 1905, instance is in a story about a New York department store worker annoyed by the operator of her boarding house:

"Well, that gets my goat," gasped Alice when we recovered speech. "The nerve of her!"

The second, 28 Oct 1905, instance, in a story about US navy sailors on shore leave, may have been a expression of good-humored annoyance...or it may have been a simple expression of amusement:

 "He was there, Patrick!" cried Shorty shrilly.

 "But I'd forgotten it was the Oklahoma — that was long before I knew you. You were there, too, then? You were ashore?"

 "Ha! ha!" He slapped Patrick on the shoulder and lay back grinning at me. "If that don't get my goat!"

In the third, 18 Nov 1905, instance the reporter commenting on a British-US boxing match uses 'got his goat' to signify a boxer's "stage fright", that is, his unwillingness to fight:

American Couldn't Fight and Had Stage Fright....

....

 ...I think the crowd got his goat, or the idea of fighting — one or the other — because he did not say boo and sat down like a mope.

To further support the theory that 'get [somebody's] goat' originated in boxing, an 1897 account (The Brooklyn Daily Eagle, Brooklyn, New York, 21 Mar, p. 7; paywalled), while it does not use the phrase, does conspicuously associate the goat with a boxer's verve, loss of which latter results almost invariably in the loss of the fight. Complicating the support of the boxing origin, however, is that this account references the goats of Harlem, thus lending support also to the origin story offered by the reporter in 1900:

 Siddons had a hop, skip and a jump which savored much of the festive goat jumping from rock to rock in the palmy days of Harlem.

As is made clear in the story, the boxing match described was held up to ridicule and distaste, not only for the part played by the capering Siddons, but also for the part played by his opponent. Both contestants had been unmanned by management that promoted a draw.

Mention should be made of another sense of 'goat', "a foolish or contemptible person" (OED, attested from around 1616 to 2012; paywalled). Use and awareness of that meaning of 'goat' may have affected the later semantic development of the phrase 'got [somebody's] goat' into "got strong influence over [somebody by angering, irritating or exasperating]". An example of the "contemptible person" meaning is this from p. 4 of The Wheeling Daily Register (Wheeling, West Virginia), 26 Sep 1877 (paywalled):

Doc. Todd...says that he has got his goats at the Ætna mill — as he calls those sturdy sons of toil — fixed all right now. He wants people to understand that he carries the Ætna mill votes in his vest pocket. To say the least of it, it is not a very charitable way of talking about an honorable, honest, upright, hardworking class of people who so nobly toil for their living.