My T is an MD (psychiatrist) which is good because he also manages my medication. It doesn't change who he is or how he works. I'm a PhD and many of my friends are, too, and so I am very much aware of the fact that a doctorate changes nothing (except your salary if you work in academia).

Neither of the Ts I saw before current T was a doctor, and neither of them told me a single thing about themselves. I was not interested to know, either. Current T has disclosed a very little, after two years, but I don't think I would want to know whether he has a family, for instance. In any case the level of disclosure doesn't seem to have anything to do with whether they have a doctorate or not.