My therapist is a psychiatrist so it makes sense to me that he refers to his patients as such. That being said, he never addresses or refers to me at all, or calls me anything to my face, because there's never anybody else present in the room so there is no need for that - my language works differently from English in that respect, and even when we do speak English, that rule holds. (I have never called him anything to his face either.) But he has occasionally said things about "other patients", so that seems to be his preferred term.

A T I saw previously worked for an organisation where they referred to the clients/patients with a word that means, roughly, "somebody who tells you things in confidence". There is no exact equivalent in English. That word is more commonly used in a religious context, such as pastoral care (not confession, though) which made me feel a little uneasy, but other than that it's a pretty good term, I think.