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A question on ELU gives some background on the use of looney. Etymonline's entry for bin has "receptacle", "all from L.L. benna 'cart,' M.L. benna 'basket.'"

Loony bin seems to be analogous to police box or paddy wagon. I'm not sure if the "insane" were literally kept in boxes or carted up as the definition implies. If not, why was bin chosen?

Was bin used in similar phrases or contexts, or was it just a unique construction that caught on?

A bin is an enclosed place for storage, such as a coal bin.

From Merriam-Webster:

Etymology:Middle English binne, from Old English binn, binne manger, basket,
probably of Celtic origin; akin to Gaulish benna
two-wheeled cart with a wicker
body; akin to Greek phatn* manger, Old English
bindan to bind * more at BIND

: a box, frame, crib, or enclosed place used for
storage coal bin *apple bin* grain bin

The OED says the receptacle meaning is used "in the
most diverse senses" and gives examples storing grain, straw on a farm-yard, a partition in a barn, manger for animals in a barn, and a receptacle for storage of food ("corn, meal, bread,
fruit") and later other things like dust and coal.

Interestingly, the following was only added to the OED as a draft addition in 1997 and first quoted in 1972:

Any receptacle for holding rubbish or waste,
esp. waste paper; a waste-bin.

1972 T. Stoppard Jumpers i. 23 Crouch enters
from the Kitchen, carrying a bin of rubbish and
several empty champagne bottles.

Online Etymology Dictionary reports that loony bin is less than 100 years old:

Slang loony bin "insane asylum" is from 1919.

However, a Google Books search finds one instance from a decade earlier. From P.G. Wodehouse, Mike and Psmith (1909):

“Nothing that happens in this loony bin,” said Psmith, “has power to surprise me now. There was a time when I might have thought it a little unusual to have to leave the house through a canvas chute at one o'clock in the morning, but I suppose it's quite a regular thing here. Old school tradition, etc. ..."

Wodehouse also accounts for the second-earliest Google Books match, albeit in the variant form "looney-bin." From P.G. Wodehouse, Indiscretions of Archie (1921):

Archie proceeded on his way, still mystified. Then he reflected that artists as a class were all pretty weird and rummy and talked more or less consistently through their hats. You couldn't ever take an artist's opinion on a picture. Nine out of ten of them had views on Art which would have admitted them to any looney-bin, and no questions asked. He had met several of the species who absolutely raved over things which any reasonable chappie would decline to be found dead in a ditch with.

I haven't been able to identify the 1919 occurrence that Etymonline has in mind. However, Eric Partridge, A Dictionary of Slang and Unconventional English (1937) has this entry for the phrase:

loony-bin. A lunatic asylum : Cockneys' : from ca. 1890. Ex loony, [definition] 2 [namely, "a fool, a lunatic"].

In the term loony bin, bin seems to refer simply to a holding-chamber, box, or other container. Partridge may be right that the phrase "loony-bin" goes back to the late nineteenth century in England, but he provides no instances of it in print from that period or later. Wodehouse may be the person most responsible for its spread to general usage throughout the English-speaking world.