It's the name that the country uses in its official communications in English, and expresses the desire to see the same name in English communications addressed to it.
They’re essentially throwing an identity tantrum on a global scale, while tip-toeing towards nationalism internally.
This would be a laughable matter, were it not a sign of what’s coming.
In a world where we are asked to refer to people by their chosen names and pronouns rather than what they were asigned in the past, why should we not refer to a country by its chosen name? Especially if it's going through the proper channels to promulgate its name change?
Ignoring the political implications of the question for a second, if nothing else when people ask to be referred to by a different name, their chosen name is usually at least reproducible on the keyboards in common use. Most people in the US would have absolutely no idea how to type Ü, let alone İ. In addition, English speakers don't have [y], [c], or word-final [e] in their phoneme inventory.
Yeah, I fully expect people without easy access to u with umlaut to just write u (regardless of case), and those without easy access to capital dotted i, to just write a dotless capital I. And the pronunciation will be brutal, at least for the first few decades, but change is a process.
So being forced to call anyone by what they prefer is cool and encouraged, except when it comes to a country?