I'm wondering now if Azumi actually was originally from this world, meaning she was originally already born as a girl. I read a light novel once with a similar premise.
Nope. A transgender is a person that is born in one gender and through a process becomes another.
This dude was born into this world as a female. His mind on the other hand comes from a male body. So gender & sexual identity is a lot more nuanced that what you are thinking.
So I would say she is more cisgender that is romantically fluid.
That is she is physically a female with female parts(cisgender) ; but due to mind swap of boy mind into a female mind his sexuality isn't solely being determined by his gender so he is having feelings of sexual attraction toward females and also of males(romantically fluid).
So, I take it that she was born in the fantasy world, reincarnated in the normal world, then reincarnated again back in the fantasy world thanks to her wish. Well, that's one way to fix the romance.
So, Azumi is a princess in the isekai world, then reborn to our world as a boy, and transport back to her original world. I guess... I'm confused here.