Do students and faculty comprise any State University in the US?
No, there's nothing like Oxford's or Cambridge's legal structure in the United States, where students and faculty are members of the University.

For historical interest, the first university -- University of Bologna - 1088 -- was a universitas scholarium, i.e. a "university of scholars (students)".  It was a collective that had bargaining rights with the city and hired and paid their professors -- paying them after each lecture. (!)