I don't think they expect or want clients to not care about them. The idea is probably more to keep the focus on the client's issues, keep those as the topic of interactions. I am sure most appreciate compassion from clients unless it is done excessively and used as distraction. Then there are those that actually make therapy more about themselves and infuse the client's situation with their own problems - I had one of those and it was not not helpful but potentially harmful.

I cared about my last therapist as I liked him a lot, but never felt compelled to use my T time to discuss his challenges beyond what was in the context of what I wanted to address for myself. He talked a lot about himself and sometimes I asked to tune it down. He once had a difficult, chronic health issue that he shared with me and I did express good wishes for his recovery several times, which he appreciated, but that was about it. Perhaps wanting to care too much is a bit similar to codependency in relationships where people sacrifice themselves and care on the surface but at the same time neglect own issues and use caring for others as a substitute for self-care?