It is likely an affectionate nickname, not a formal title. Probably just a way of assuring her "patients" that they are in good hands, seeing as she is clearly well versed in the science of medicine.
Wherever Professor Oak got his.
I'm pretty sure there aren't many universities in post-apocolyptia, so I wouldn't count on too many people having degrees.
- There's one in Los Angeles, but she probably didn't go to it. The Institute may count though...
Don't let Dr. Li touch you, she's not a real doctor! Really the problem is that Fallout 3 keeps forgetting the war was 200 years ago instead of 20 - like how Dr. Pinkerton's story about leading survivors of the NRI to found Rivet City is told in such a way that it sounds like they did so not long after the bombs dropped.
Pretty sure if you know enough about science or medicine in fallout, the title falls into your lap.
- True. Also, look at the Doctor in Bigtown, Red? Even if Bigtown or somewhere nearby had a university, there's no way she could be old enoughto have any kind of degree, especially not a doctrate.