When I was diagnosed several years ago with gender dysphoria the therapist didn't give me a letter rather she communicated the diagnosis to my doctor who prescribes hormone therapy. Shouldn't she have given the letter to me directly?
Doctors do all sorts of odd things, but its possible its some misguided attempt at reducing forgery. Or just a power trip, who knows.
Personally I get copies of all my medical records as you never know when they are going to be useful.
My therapist sent one directly to my surgeon and sent me the same copy. I think it may depend on what kind of healthcare system you're in. I know a few places do everything in house now as far as therapy and HRT go (with some adding in various surgeries). In that situation, there'd be no "need" for you to have a copy unless you wanted it.
Have you asked for it? I think the bigger concern would be if she was withholding it from you after you'd requested it.
I never got one from the therapist as I did informed consent but the letter from my doctor's office listed my diagnosis.