My understanding is that it is not good for T's to accept presents because it might influence the relationship. Most T's seem to approach this boundary with strict "no gifts" policy. This way they don't have to get muddled into trying to figure out each situation.

That being said, I know my T accepts some gifts. I have given him artwork, and he puts it in a special place in the room. If I wanted it back he would give it back, I believe. He got some fake flowers from someone once, and they were in the room for a few weeks, and then they disappeared. I don't know if he would receive a bigger gift.

It sounds like your T has very strict boundaries. Some T's do. The fact that your T admonished himself for revealing personal information, even a little bit, shows how strict their boundaries are. It is a general rule that T's shouldn't talk about themselves, but sometimes it leaks out a bit, here and there, and for most T's that's fine because it's part of the process.

Maybe you could talk about how the experience felt to you next session, and discuss their strict boundaries. There could be more you don't know about.