That was in his office, in his home, he'd like to be called Dr. Marvin.

Honestly he's just being a dick.
Green Goblin is great!

It was this, and it is pretty simple. In the office, it would be okay because that was probably where Leo felt he would see him only. He never expected Bob to be in his home. I guess he wanted to keep his home distant from Bob. There is nothing wrong with that. In fact, there are some doctors who choose to be formal with their patients even in the office.

I now see only a psychiatrist (been on Prozac since 2015) and have called him Dr. (Last name).

leo is a pretentious dude.

I have an aquaintance that I met in a casual setting and have never known her through her professional capacity as a psychiatrist. She signs her emails to me with Dr. (her full name). It could be that she writes to her patients using that email, but it's unlikely as it's not a work email. I never bothered to ask her.
On the other hand, I know someone else I met through her profession as a Chaplain in a hospital. She has a doctorate and she introduced herself to me by her first name and last name and signs her emails using just her first name. I email her back using "Dr. (first name)" as a sign of respect.
All in all, I think it's an individual choice made on a case-by-case basis. In the case of Bob and Dr. Leo Marvin, I think the latter was just trying to keep the lines of the relationship clear and he was annoyed that Bob not only barged in on their vacation but his family was so welcoming toward him.

Does this rule only apply to psychiatrists? I see both a case manager and therapist at my mental health clinic and they both allow me to call them by their first name, but they don't have doctoral titles.

To call somebody by their first name kind of implies a friendship of sorts. Leo wanted to maintain the patient/Dr relationship with Bob.