Depends on the animal.
My fish don't mind. They like everyone. They go happy-spaz when anyone goes near the tank.
When I was living at my parents place, they had a cat that loved men. Weird cat, was normally a bit standoffish until I started T. 2 weeks later and the cat was all over me.
I didn't change my appearance at all - only the T. My arrival pre-dated the cat's arrival by several months - so it wasn't just the cat getting used to me.
The other cats that sucked up to everyone didn't change behaviour at all, only the one with a gender preference.
It wasn't just me that noticed it, my folks remarked that the cat seemed to like me a lot all of a sudden for no reason

Considering their dogs didn't start barking when I arrived there last time, I don't think animals think you're another person. To them, you're still you - you just smell a little bit different after HRT (but not different enough to smell like an entirely different person).