Is Emmett Brown an actual doctor? It doesn't look like it's shown in the film and 'Doc' or Marty is an unreliable source himself.
Ironically, right then, Dean Wooster stops by with two other men named Cooper and Mintz. They tell Doc that, because he is working for them at the University (as a professor of physics!), he has to participate in one of three projects--working on the Edsel car, working on a chemical warfare thing ("They'll name the final chemical agent after you," notes Cooper. "'Agent Brown.'"), and working on something called Xerox, which Doc pronounces "X-rox." Doc, however, has no interest in any of these projects (even though Marty tries to steer him towards the Xerox one!) and refuses to do them. The Dean then threatens him that his relationship with Jill will be damaged unless he cooperates with them.
Typically, you have to have a doctorates in the field you teach, showing Doc to have a Ph.D. in Physics.*
Modern day universities cheat by having non-doctorate lecturers, not professors, a practice not used back in the day.
This scene happens in 1955 after Marty got the courier letter at the end of BTTF 2, and tracks down 1955 Doc.
This also changes how the Doc/Clara romance is portrayed. It initially had Doc in a relationship with the Dean's daughter, who turned out to only be using him on her Dad's behalf, contrasting the future (past) relationship he has with Clara.
There is no evidence in the Back to the Future movie franchise demonstrating that "Doc" Emmett Brown had any kind of doctorate - almost as though "Doc" is a nickname. The likely reason for that is "he could never stay on one subject long enough to write, much less defend, a doctoral dissertation".
The thing is, it's just a movie. If the filmmakers tell us he's a doctor, we just kind of have to believe them. Otherwise they have to shoehorn in some framed doctorate on the wall, or something like that. Is there any evidence that Marty's name really is Marty? Maybe it's Paul, and everyone just calls him Marty. They certainly don't show his birth certificate or anything. If filmmakers had to give evidence for every aspect of the setup of the story, the story itself would never get started.
So in a way, the answer is yes. The evidence is that the character is called "Doctor Emmett Brown". He's certainly clever enough.