In human languages, each culture develops its own name for each different culture they interact with. That name is a separate concept to how the other culture calls themselves. Sometimes, the two names are relatively similar (France / France, though pronounced slightly differently), other times its completely different (Germany / Deutschland, Hungarian / Magyarország). This is a simple fact about how human languages work. The country called Turkey in English calls themselves Türkiye in their own language. They have asked certain international bodies to refer to them by that name as well, and those bodies have of course accepted. But those bodies do not "speak English", and there isn't any expectation that speakers of English should change their language as well.

Now, there is nothing wrong with calling it Türkiye in your own everyday speech, but it is not any kind of sign of good manners - it's simply a matter of personal taste. You'd still probably call their language "Turkish", while they call it "Türkçe", and the people living there "Turks" or "Turkish people", when they call themselves "Türkler", and even if you didn't you'd probably use your own grammar to conjugate these words as necessary etc.