I don't know that it is an obsession; I think that would take the same person posting all the same sorts of threads?
I loved the look of my therapist's hands; my favorite picture of my mother and me (she died when I was 3) is of her holding me on the way to my being baptized when I was about 6 months old; my father took the picture and I love the look of my mother's arm/hand under me, holding me; so strong and "safe".
My T wore a wedding ring and I did not wear mine because it was slightly small and would "bother" me if I wore it too long. But, my T's ring looked well on her slim, light brown hand, and I wanted to be like her (calm, quiet, self-possessed, engaged with the "other", a good listener, etc.) and started wearing my wedding ring on therapy session days, mirroring her.
Well, one day at work I was covering the front desk for the receptionist and the bookkeeper came to relieve me and noticed my ring and exclaimed, "Perna, you're wearing a wedding ring, I've never seen that?" and I almost replied, "That's because I only wear it on Fridays!"
My almost reply cracked me up so I told the bookkeeper the whole story, what I'd almost said and why I wore the ring on Fridays, etc. and then, when I went to therapy, I told my therapist too! We both enjoyed it and laughed and she said "someone would think you were part of a weird cult!" (The WOROFO? Wearers of Rings on Fridays only? :-)
The experience made clear to me and made me acknowledge completely to myself, why I was wearing the ring; it was a wonderful sharing experience with a friend at work and even more, with my therapist; and there was the knowledge that the person at work had "noticed" when it was such a small thing that I barely noticed/didn't acknowledge completely to myself.
Who knows who has a good "story" about their therapist's hair and what noticing/not noticing that or their therapist's eye color has done for them? Some people think they look at other people, look them in the eye but then they can't tell you the eye color? What's with that? That would be an aha moment to me, I look or try to look people in the eye but it's mostly just a glancing blow

not a real "look".
I saw a picture of my husband's mother back 20 years ago, (she died before I could meet her) taken 50+ years ago and immediately saw her long lashes and how her eyes looked like my husband's (actually, how my husband's look like hers :-) and immediately went and looked more closely at my husband's eyes and saw his beautiful long lashes for the "first" time! Yes, I knew he had lovely blue eyes but had never looked at the components or thought about what it "meant" (the long lashes help in the "beauty" for me, I then saw).
Anything can be used in therapy to help us. If you aren't into hair, eyes, hands :-) or physical prompts, maybe it is sound of voice or mannerisms that you aren't even aware that you are aware of? There has to be something to connect one to another person, to keep us engaged with that person. The person might remind us in some way of another (transference) or things we like (look of their hands, clothes they wear, how neat, casual or sloppy they appear to be, how they respond to us or what we have to say), who knows what it is until it is discovered?