Graduate students (who teach, do research, and some administrative tasks for the university) occasionally do. What American schools usually have is a “student government” which is approximately 90% roleplaying as elected officials, and the remaining 10% is deciding which banquets they should host themselves to spend the small budget the university gives them.

Not undergraduates. The "student union" in this context is 100% a social entity.

Graduate students sometimes are part of unions, but usually only if they're also employed by the university (somebody paying full tuition for an MBA probably isn't in a union, but a doctoral student teaching or doing research might be).

Undergrads doing part-time work at the university to pay bills (dining hall, bookstore, etc) could be, in theory, but probably aren't.

Even if this student isn't a member, the local graduate student union (https://www.uaw4121.org/) would probably be a useful ally. All TAs are in the bargaining unit, and UW CSE has _a lot_ of undergrad TAs, so I wouldn't even be surprised if this student is a present (or former) member.