UNC alum here. It’s definitely an old school way to answer a question like:
Q: “what college are you going to go to?”
A: “Chapel Hill”
But people did say that back in the day. I don’t hear it much any more and I would, living in NC and having college age kids.
UNC itself is really big on “Carolina”. They talk about “The Carolina Way” a lot.
There is plenty of officially licensed UNC merch and some UNC-Chapel Hill merch along with the “Carolina” and “Tar Heel” (two words, both capitalized) merch. I do love a local non-official shirt that says “Chapel Hill vs All Y’all” but that is the rare example of college related “Chapel Hill” merch. (There may be some town of Chapel Hill merch with “Chapel Hill” on it.)
No one, however, EVER says just “chapel”. That is like nails on a chalkboard wrong.