Omitting indefinite article has two basic effects in my opinion. In examples like this, I feel like it "adjectifies" the noun more. Egy also means "one" in Hungarian so whether you want it or not, using egy means you kinda count the noun too. So, using egy may be a bit more like "is he one instance of a student" while omitting it is a bit more like "does he meet the criteria of studentness?"
Another even more common effect is generalizing the noun. Just like you would use "some blablablas" in English. Not that in Hungarian, quantifiers are always used with singular, which makes singular noun without any articles the most suitable solution for talking about indefinite amount. It works like uncountable nouns in English - the only exception that it works for every noun.
Ok here's a situation that doesn't translate well. The sentence seems to imply that someone in a school is acting like a teacher, but it's assumed he/she is a student. Someone comes along and sees the student acting like a teacher and asks, "Is he a student or the teacher?"
That's what I put, and I stand by it. Is he a student or teacher is also right, but when it marked me "wrong" it said I should have said "Is he a student or a teacher." That seems unnatural to me.
WHY CAN'T I USE GENDER NEUTRAL PRONOUNS IN ENGLISH??? Ok, It's out of my system, but "Ő" literally translated to singular 'they'. I know that they don't like this as an answer because there is a different word for the plural 'they' in Hungarian, but it is perfectly grammatically correct to say "are they a student or teacher"... I'll just keep reporting anything that doesn't accept 'they' for 'Ő'...