Thanks for all your responses, I feel reassured about my T, especially as Echos said it is not a requirement. Because my other T's had all asked for my details, I assumed it was a kind of requirement. I do respect them for taking my details, but I am reassured that my current T is not being necessarily unprofessional by not taking details.

I think this concern is related to wider concerns about whether I have chosen a good T. How can we possibly know at the start whether they are a good T? I think only time will tell.