I had a session similar to this after about a year of therapy, though my T said he usually does it in the first or second session. Due to circumstances we couldn't do it then.

He asked me about my family history as well as my own history, so things like 'which school did you go to' and so on.
I think it's good for a T to have a general overview of the life and family of a client. Your T can put it all on one or two sheets of paper, has all the information there. I don't think it's meant as a comprehensive list of everything, just a general thing.

If it were me, I'd probably not tell a new T everything either. But I'd probably mention that there's things that I'd rather not talk about and would say 'I don't want to talk about X right now' rather than just completely step around the topics. That's because if I'd feel that uncomfortable around a certain issue, I'd probably at some point want to discuss it in therapy. Maybe not right now, but at some point in time. And the T wouldn't know there was an issue there if I'd never mention it, and then at some point it becomes weird to even bring certain things up.
I don't really have any input on how to avoid topics completely, since it will probably more be questions from your T that you answer, and you can't really avoid hearing certain questions. In my opinion, it'd be best to just be upfront if you are uncomfortable with something, but still say that.