Is the villain in Disney's Princess and the Frog a doctor, and what is he a doctor of?

First off, it's notable that Dr. Facilier is actually the only one in the entire movie that uses the name, and the rest just calls him the 'Shadow Man'. Also, there is no explanation for his name in his fictional character history (as far as I found, at least).

Second, while he is most certainly a bokor, sometimes known as a (voodoo) witch doctor, this is by no means sufficient explanation for his name.

In an interview with the directors John Musker & Ron Clements, Muskers has said the following:

"Dr. Facilier was originally Dr. Duvalier but we didn’t want to confuse him with the ruler of Haiti with that same name."

They changed his name to avoid confusion, but it appears Dr. Facilier was still heavily inspired by this 'ruler' of Haiti [called a dictator by others], François 'Papa Doc' Duvalier. Papa Doc also stylised himself after the Vodou loa Baron Samedi, and revived Vodou traditions on Haiti. Baron Samedi is characterized by the top hat, black tail coat etc., just like the Shadow Man wears. Duvalier actually studied medicine and worked as a physician, and was called 'Papa Doc' by his patients.

I would say that the 'Dr.'-part in 'Dr. Facilier' is mostly based on Papa Doc, but it also works with the whole 'witch doctor'-thing. I'm aware this is not definitive proof, but based on the available evidence, I can't think of anything else that makes sense.