I sometimes thought about how come that some of us need/want that, some don't care and some (me included) even fear to meet their T in RL. Would surely be interested to read others take on that.

However, regarding your case- the only Q that come to my mind is: does your T share anything from her personal life with you? Personally I'm quite bothered by the "tabula rasa" approach of some T's. I think it can help the client to know stg about their T. I think even I would become obsessed with my T if he didn't tell me anything about himself. We (meaning I) do some small talk in the beginning of the session or when things get tough.
Anyway, just a thought.