Notice that he does a neat little switch there. He talks about Princeton, then he says "As a professor", then he talks about Princeton again. He's a professor at George Mason. I think anyone can figure out why he used Princeton rather than the place he works without actually stating the name of his employer.

It's odd, for sure, but perhaps he just uses Princeton as a relatable example. Most of us have never heard of George Mason.

I went to an Ivy League school, and while an ID was needed for some resources (mainly the libraries and gyms), no one would have batted an eye if you went into a decently sized lecture.

Sure, but the big lectures (mostly) aren't the draw--it's the smaller seminars. An extra person would definitely stick out in a room of 10!