It depends on the universities. Yeah for most public universities, the grad students are directly funded by the professor's research. In many other universities, grad students are funded through the department, as in they are guaranteed a salary; the salary would come from the professor (or other fellowships the student got themselves) if they a research assistant, and it would come from the university if they are a teaching assistant. They are guaranteed funding because if the student doesn't get research funding, then the department sticks them in a TA position so that they get a salary.

Isn't it mainly TAs who want unionization? At least in the physical sciences, RAs are indeed paid with grant money.