I understand the weird feeling. Early on with my current T, during a session she revealed a personal issue. It was very strange at first and I asked her why she would share that info and that I felt awkward knowing something so personal. She told me that it was ok to feel awkward. She told me that she shared it with me because she felt I needed to know something that made her vulnerable so I would trust her when I felt vulnerable.
Perhaps your T is trying to establish a rapport with you as a means to demonstrate a different way to deal with others or as a way to learn where your boundries are. If you don't mention how you feel, how would they know your thoughts and feelings?