“He’s been a student for going on five years.” Is this sentence correct? Can “going on five years” mean almost five years?

Yes, that’s exactly correct :)

It kinda makes sense if you drop the "for," but I've never seen it used in this structure. To me, using the phrase "going on xyz time" is usually a response to "How long have you been doing abc?"

Yes, it can absolutely mean that: https://dictionary.cambridge.org/dictionary/english/going-on

Yes. And yes.

It's almost correct: "He's been a student, going on five years \[now\]." Don't need *for*. >Can “going on five years” mean almost five years? It doesn't mean anything else.

Where I live, people say "for", I live in Texas, must be a regional variant then.

Yes, it's correct but "he's been a student for almost five years" sounds a little more natural in my opinion.

Yes, it's correct. It typically means "almost five years" or "approaching five years". It's an informal way of expressing that something is nearing a specific duration of time.

I feel that “He’s going on five years of being a student” is better. Typically the saying “going on” comes before the subject.