Interestingly though, en mus in the sense of a computer mouse has a regular plural (musar), while the animal has the irregular plural möss. I think this mirrors some English use too, but I'm not sure.
I disagree, I say möss about computer mice. The only time I'd use the plural musar is for a kind of word we're not teaching here.
I googled and got 20,400 for mine and 7,100 for yours so both are used but möss is more common. – SAOL only lists möss though.
You're right about English, they have approximately the same situation. A fairly large minority use computer mouses.
"A mouse" is just one mouse, "mice" is more than one mouse. "A mice" is not proper English. You just say "mice" without the A.
Wiktionary says it's en sparv ('a sparrow') in Swedish, but it sounds more like en mes ('a tit' in English) (which in turn seems to be mees in Dutch). Anyway sparrows and tits are small birds that are easily confused by most people.
Compare "An egg" / "a blue egg": whether it is a or an depends only on the sound that comes after the a/an. If it is a vowel (a ee oo e o...) it is "an", if it is a consonant (b, c, j...) it is "a".)
So to make it short: there is no difference in meaning between a and an, the placement only depends on the pronunciation in english, therefore the only way to know which one "en" is going to be is to learn the rules in english.