I'm not a therapist. ExDH was clinical psychologist and I remember having this discussion with him. Apparently the role of a psychotherapist / psychologist, as first mooted by Freud but since further developed obviously, is that they are a sort of neutral sounding board, a gentle questioner, this is apparently the best way to explore the patient's "neuroses" (or whatever the modern correct clinical word is!).
Apparently insisting on a therapist with certain sympathetic beliefs can compromise the patient's ability to fearlessly explore their issues and can be used as a way of avoidance on the part of the patient, or a way to subconsciously reinforce existing "neuroses".
He gave me an example of a conservative religious Christian father who fears his son is gay, so ensures he has a therapist who also shares fundamentalist Christian views. Or another example, a patient with anorexia whose therapist knows a lot about eating disorders and is very skilled but still not fully psychologically recovered from her own anorexia.
Something about bias and projection, i can't remember everything he said, just his emphasis on neutrality (although he used a different fancier word lol) in the approach to the therapy. 'Neutrality' is how the patient gets the best results.
I suppose a more current example would be whether 'trans affirming' therapists are the best bet for teens who may feel gender dysphoric but not sure if they are trans or not.
Now I'm not a psychologist and I know there's a massive difference between clinical psychology and 'therapy' which is usually something quite different. And of course there must be loads of clients who would prefer a therapist of a particular gender, sexuality, religion etc. Plus there's also a need for trauma-informed therapy where say a rape survivor wanted a woman therapist. I don't know all the ins and outs, just that in clinical psychology, this neautrality thing, in the approach to the patiernt, is key, the therapist should not be forming the treatment through the framework of their own beliefs to the extent that is possible of course.
Hopefully an actual clinician will shortly be along to explain it better!