I think Ts do get attached to us too. I think attachment has a more negative connotation, at least in this forum, than caring, but attachment means that you feel close to a person. It's a good thing unless you get overly attached, which could mean that you're neglecting your real life relationships.

Caring about someone seems to me to be always positive. My T has told me she cares deeply for me but I don't think she would say she feels attached to me. There are attachment styles; you can find information about them online. The goal would be to have a secure attachment to your T rather than an insecure one. In that use of the term, attachment is positive.

I'm sure you'll get more explanations, probably better than mine.