I don't know if we can call it "obsessed" since that has such a negative connotation.
For me personally, I thought about my T and every session all the time. I didn't find PC until I lost her and started with a new T. I wish I had found this place years ago! Therapy would have been different with old T! This place has really opened my eyes and helped me realize I'm not alone, many others have gone through the same thing, my feelings are "the norm" and I'm not some sort of freak of nature. I'm learning so much about therapy and how to use it to get more out of it. I didn't know squat before. It empowers me to be more in charge of my own therapy rather than expecting my T to be in charge.
If you aren't thinking about therapy in between sessions, then how do you expect it to help make the changes you want to make in yourself?
If the thinking about it too much turns into something negative in your life and causes more harm than good, then maybe you can call it "obsession".
Just my 2 cents, thoughts, ramblings...