The government gets no extra authority from its ownership of the forum. But it will be able to bar discussions about, say, Shakespeare, or discussions conducted by non-university students. On the other hand, one could argue that the government shouldn't be able to designate fora in a viewpoint-based way. We strongly suspect that courts would not allow such viewpoint-based designations, but the answer isn't completely clear.
Public fora are categories of government property.
It's simply private property, and not a forum of any sort.