Gender dysphoria can be medically studied, is not an ideology and is not a disorder. Hope this clears things up.
But identity as a whole is a very murky thing - if you ask me it's largely an adaptive abstraction that our minds invent.
The purpose of said adaption is to adopt a role which functions within the tribe/society for purposes of survival.
I think we way over-simplify the whole thing by making it about gender and gender roles.
And it's that over-simplification that I would label as the ideology. Because that's what ideologies do: they take the complex ambiguities of the world and try to cram them into a simplistic box.