At least in the US, having a Ph.D. and insisting on being referred to as "Doctor" is considered a real asshole move. Usually one might address the person once as "Dr. Whatever", but they will invariably reply, "Please, call me Jane".
Even referring to someone as "Dr. Whatever" in the third person is pretty unusual, if you've ever met them. If I was to speak of my college professors right now, I certainly wouldn't use "Doctor". Maybe if they were 70 or had a Nobel or something.
(Related: In the movie Avatar, one scientist introduces himself to another scientist as "Doctor Norm Whatever". It absolutely clangs, at least to my ears.)
If you know them sure. But for a professor who I took a class with? I would always call them Professor Whatever if my only interaction with them was in a classroom context.
Maybe a generational thing? Not sure when you went to school, I went to a midsized state school 3 years ago and mostly referred to professors by their first or last name. I think this is more common in CS perhaps? I don't think my physics friends did the same thing as often though.
Related: as an anesthesiology associate professor (M.D.), I always told the residents to call me Joe. However, there were always a couple who either would not or could not do so, and addressed me for the entirety of their three-year residencies as Dr. Stirt. Diff'rent strokes
I think she asked us to call her by her first name at the beginning of the semester and it stuck (but I can't remember and I might be being rude). Honestly I'm as surprised as you are - mulling it over I'm sure I called her Prof. Dwork when talking to her, since that's how I addressed most professors I didn't well personally.
I went to University of Illinois at Chicago (not UIUC or UofC). If I worked with a professor for awhile (not in class more for research oriented things) sometimes it would be appropriate to address them by their first name but I never did.
There were only two professors I was on a first name basis with in college. One was my advisor, I also worked in his lab for two years. The other was another professor in the department. It was different in grad school. It was first names for professors you worked with. At that point, they don't know more than you do about your research.
A U.S. thing I’ve noticed is calling professors “Dr. Smith” when elsewhere (and certainly in Germany) that would almost be an insult, since it’s “Prof. Smith”.
I know a German professor with two Ph.D.s who calls himself "Professor Doktor Doktor So-and-so". He doesn't introduce himself like that, mind you, but he uses it in his .signature file (is it still called that nowadays? the text that's automatically appended to your email).
Is this a typical Harvard thing?