Hcfyk

Mir muss Lernen Spaß machen. Spielen macht mir Spaß, aber auch Lesen.
Deshalb lese ich gerne Co­mics in einer Fremdsprache oder spannende
Ge­schichten. Dabei merke ich mir die Wörter viel schneller. So habe
ich auch Schwedisch gelernt. Erst habe ich Wörter gelesen, dann
Sätze. Irgendwann konnte ich ein System erkennen.

Why "konnte" here? Why not just “kann”? Is the narrator emphasizing that he had a system at some point in the past but not anymore?

The writer describes the general conditions that make language learning easy for them. This statement is valid at the time of the text, so it uses present tense, Präsens.

Then they give an example from the past (when they were learning Swedish), which therefore uses the tenses of the past, i.e. Perfekt and Präteritum.

At first, the writer explains his/her methodology for language learning.

From the 5th sentence on, the writer says that these general conditions helped him/her to learn Swedish ("That's how I learned Swedish").

During this period, the writer could recognize this learning system ("At some point I could recognize a system"). That's an action that happened in the past (therefore, the use of konnte is correct), but that doesn't mean that this system doesn't exist anymore.

"Konnte" in this sentence is similar to the use of "was able to" in English.
My English translation would be:

At some point, I was able to recognize a system.

Why "konnte" here? Why not just “kann”? Is the narrator emphasising that he had a system at some point in the past but not anymore?

The problem here is that you interpret "[...] konnte ich [...] erkennen." as the description of a state (seeing a system), when it actually describes an event in the past that marks the transition from one state (not seeing a system) to another (seeing a system).

Your interpretation is not wrong:

There was a time span (that is over now), when I was able to see a system.

But most German native speakers will understand it as:

There was a point in time (in the past), when I started to see a system (and I still can see it).

I don't have a source or rule, but from my intuitive understanding I would say that the latter interpretation is what comes to mind first when the topic is about identifying something or learning something.

If you wanted to express in German what you understood, I'd phrase it like this (but this might be colloqiual, not sure):

Irgendwann konnte ich mal ein System erkennen.

or more explicitely:

Bis zu einem gewissen Zeitpunkt konnte ich ein System erkennen.

I'd argue "konnte" in this context is used in colloquial speech. It is one of the weakest words in terms of accuracy and if applied like this, adds some uncertainty to the statement. In essence, the reader cannot be sure to which extent the writer grasps the rules, ins and outs of the so-called system, but they do recognize a pattern.

When talking to other people that way you put yourself in a safe, defensible position. You imply to know/understand things, but don't proclaim to be an expert.

I am German, short answer:

"konnte" in this context is the simple past of "kann" of the verb können. As you know German knows much more times and cases than English.