There are a lot of inconsistencies in Frasier's academic history too. They say that he's an MD/PhD, but started in private practice at 31.
Frasier mentioned his father was a professor in
Cheers, although when he later got his own show he did address the inconsistency stating that he was insecure and something to the effect that he didn't want people to know he had a blue-collar dad.
He met Diane when she was at an inpatient facility. I believe it even stated she was his patient (this is based on a memory from years ago).
Frasier as a whole didn't depict the way most psychiatrists practice these days or even how most practiced when the show came out. With such prestigious institutions, and Frasier knowing people he alleged were top people in the field (usually are in an academic institituion and/or doing research), I wouldn't expect him to do the type of work he did, though it's not out of the question or completely unrealistic as I do know some doctors that trained at top places and do things like private practice.
The show, per the writers and producers, was focused on neuroticism and making sure that every character had something they wanted but were never able to get at the end of the day. Frasier-never being able to find someone he felt secure enough with to settle down, Niles wanting Daphne but never being able to confess his love for her, and Martin being driven batty by his two supercilious and narcisstic sons. The emotional ammo was supposed to come from the characters going through their neuroticisms.
This perfectly explains why the show jumped the shark when Niles finally ended up getting Daphne....
Another very unrealstic detail was Niles fainted at the sight of his own blood in one episode. He wouldn't have been able to make through medschool or his primary care portion of residency with such as problem.
If a psychaitrist existed with a personality similar to that of Frasier or Niles I'd expect them not to work in community care (because they're too narcissistic) and either be in academia or research at a place where they wanted an elite name attached to it, or doing very expensive private practice while thinking that they were somehow perhaps better doctors because their patients had wealth.