My new T is sort of like this... perhaps to a lesser degree. She self discloses a lot more than my old T, but her modality is relationship and attachment based so it makes some sense that she's trying to form a bond or connection with me. She doesn't text me though unless it's about appointments/scheduling, although she has told me that I can text her for things other than that if needed (I haven't). I also don't think she'd give me anything for my bday, but I don't know. I'm an attachment-phobe, but she's letting me take it at my own pace and seems to accept that I'm going to fight it every step of the way... perhaps you could tell your T to tone it down a little, that you're not "there" yet?

I guess my thought is, what modality does she practice? Maybe it's her style. That said -- if it makes you uncomfortable then you should definitely bring it up.