This is what my current T is doing all the time in session. He is very interactive but not overwhelmingly so, just has a way of relating his own experiences/thoughts to mine and it works well to keep me relaxed and present. The way we interact is not asking intrusive questions out of the blue but expanding the ongoing conversation topic. For example, I think it is possible to learn a lot about someone from their opinions and reactions and it does not necessarily require knowing factual information about their life.

My former T (a psychoanalyst) was quite different, he rarely shared anything about himself in session and his reactions to what I shared were typically quite short. Sometimes would not just sit there looking at me and make some superficial comments. I always felt quite uncomfortable in sessions like that and usually told him, then he would become more interactive but far from current T.

I think it is perfectly fine to tell the therapist what we do and do not find helpful in their style.