Yes, but you have to know a little more about the student. You can say an "undergraduate" if the student doesn't have a Bachelor's degree yet, or a "graduate" if the student is studying for a Master's Degree. If the student is studying for a Doctorate, you could say "doctoral candidate" or "PhD candidate". Similarly, you could refer to the student by his/her year in university: freshman, sophomore, junior, senior. This also applies to high schools (at least in American high schools), so it's not totally clear without knowing the situation.

The ones i can think of the top of my head is "attendees" - as in the people who attend a school, and pupils (i don't hear my American friends using this one very often)

In the UK you might hear someone described as "reading" - for example: Stephen is reading PPE at Oxford.

intéresting

Often “student” wouldn’t be needed. “He is at college” “He is at Harvard” (“student” is implied in both cases) “He’s at Harvard.” (To sound more natural)