> Exception is when the noun is referring to its concept in the abstract

You need an article here.

If there's only one exception, it should be, "The exception is when...", and if there are multiple exceptions, say, "An exception is when..."

In this case, it's grammatical, just for a different reason. You can sometimes drop the leading article or pronoun of a clause -- don't know why, just happens sometimes.

("I" don't know why, "it" just happens sometimes.)

Or for instance, if you're viewing a house on the market: "Kitchen is to the left, sitting room to the right; office is upstairs." It's certainly a casual register, but it's no less grammatical than including the articles explicitly.

In the ancestor comment's "Exception is when ...", the article is implied but elided, in the same way. (I inferred "the".) You could argue whether it's effective to drop the article there, because it's ambiguous in the way you noted, but it's still grammatically valid. (Chomsky's "colorless green ideas" is the usual example where semantic sense diverges from grammatical validity.)

This is a good defense of it, but as a native English speaker I still found the sentence jarring.

It’s possible that my sensitivity to this kind of error (or causal register, if you prefer) is due to experience learning other languages that don’t have articles like those in English. I doubt it, though. Many others probably also find sentences stripped of articles as jarring, and that’s useful for anyone learning English to know.

As I learner, I generally want it to be understood by speakers of the language as easily and with as little friction as possible.

Yeah, you can't always drop articles -- like I said, it usually happens at the very start of a clause, and not at all interior to one. And whether it's advisable to do this is up for debate; you make a good point about accessibility, and it's not always even natural. But it does happen, so it's worth cataloguing the existence of this phenomenon at least :)