a bit more of my thought process:
my best guess about the family situation is that she has a dead sister (hence the zombie comment from her brother.) since it looked like the family photo had two sons & a daughter, but only two of his kids came to visit (one son & one daughter, but seemingly not the daughter in the photo) (& she also apologized to the person her father mistook her for)
i think it was her being made to take the role of his daughter that then made her realize she wanted to be a woman. especially because if all it was was just a role she was forced to play by her father, i really doubt she'd then go to the effort of socially (possibly even medically) transitioning (and trust me, it does take EFFORT.)
in my mind the director is mtf. her talking ab "what if he decided he wanted to be a woman" when she saw pics of minato crossdressing was sort of a key to me. i think she was forced into a role by/for her father but eventually realized that she likes being a woman more.
also, t doesn't come in pill form. e does. (assuming that's what she was taking with her coffee)