My T says that she can only treat people if she feels a connection to them. I'm not sure how that relates to your definition of caring.
But I believe that she feels a connection to everyone she works with. The relationships may not be the same, but the connection is always there. She says that (in her 30+ years as a therapist) there have only been a few people that she couldn't feel a connection to, and she has referred them to other therapists.

So I guess I think that she, anyway, cares for everyone she works with. But I think that the quality of her caring is not the same as the caring of (for example) a family member. I think that her caring has a lot more of respectful understanding in it, and a lot less neediness.

I have had the "why can't we be friends?" conversation with her, and she says, "Yes, I wish we could be friends too. But if we had that relationship, then we couldn't have this relationship."

So, I don't know. I guess I have come to accept our therapy relationship as a kind of unique thing -- not really comparable to other relationships, but still a very real and intimate thing. And I guess that I think that yes, she is able to have that with me, and also with all the other people she works with.

But it's definitely a puzzle.

Take care,
-Far