I'm sorry it was such an awkward session after a difficult week. I would be very bothered by the way it went down. I would also really be bothered by the potential breach in confidentiality with meeting her daughter. She has no obligation to maintain confidentiality, and kids rarely think of that stuff. she may have a directive not to speak about mom's clients, but from personal experience, kids/teenagers will not understand the absolute nature of the concept of confidentiality (my best friend's mom was a psych nurse and one time my friend stopped by the office to get something from her. She saw a very prominent individual come out of mom's office, and I heard about it later that day. I was instructed not to tell anyone by my friend, but clearly the bounds of friendship excluded the confidentiality stipulation)...
It sounds like T really cares about you, and wants to be there for you when you are struggling, but it also sounds like she is blurring boundries a bit too much. I think I would have reacted like you did: offer to leave, but if T said it was ok to have the session, I would have sat through it, then gotten upset about how badly it went. If this is a trend with this T, you may want to consider either having a conversation with her around all this, or finding another T. (There's nothing wrong with asking T to reign in personal boundaries around your needs. It's not easy to do necessarily, but it's ok to say that this isn't working for you).