Do American university students belong to unions?

It's very common in the UK. The most visible part of the unions is running social activities, often bars and events, but they can also provide legal advice to their members.

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.

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.

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.