No, they are basically not-for-profits, more like what the United States more sensibly calls "private universities". But if they take government funding (directly or indirectly through subsiding enrolment fees?), they have to follow some government rules, and all major ones do it seems.
A similar confusing naming pattern appears to apply to Public schools.
In the University of Cambridge for example, all MA degree holders or higher appear to have some voting power: https://www.cam.ac.uk/about-the-university/how-the-university-and-colleges-work/governance (archive)
This adds an extra layer of difficulty for the average taxpayer to make changes to university policy, e.g. making universities publish all material with Creative Commons licenses. At most, voters could require this indirectly through the government funding requisites. It is a mess.
Not even the Open University seems to be very open!