I didn't want to disclose former T's name at first to current T because I was still in therapy with her and was feeling weirdly protective about her. But, that changed once I terminated.

With former T, I also didn't want to disclose where I lived / worked / who I hung out with / basically anything at all about my present-day life other than the bare minimum identifying-info scrubbed pieces. It became an issue to the point that she repeatedly commented on it.

With current T though, strangely I don't feel the same way -- it's really a non-issue. I feel no major urge either way and if stuff comes up, I just say it. Current T also has a form -- from the Stone Ages I think -- which asks for a lot of identifying info (including SSN!).

She really didn't seem to care though if any of the fields were left blank (the form looked like she'd made a bunch of carbon copies back in the 90s or something when she first started practicing and then never changed it).

But, apart from a bit of queasiness in terms of general data privacy stuff -- which at this point is kinda too late to worry about since both a government agency and a couple of private firms have given me free multi-year identity theft protection packages after my personal info got stolen when their systems were hacked -- I didn't really care about her knowing all this info about me. So yeah, it's weird that I had such different reactions to different Ts.