Taiwan is less recognised by the outside world than Kosovo is, and it has its own country code. Same with Palestine. Having a country code is only weakly related to being a country.
As does rfc1591
“The IANA is not in the business of deciding what is and what is not a country.”
Like, what’s even your point?
Since the de facto territories of the two states are common knowledge, you can unambiguously refer to them just by mentioning their rough position on the map. Similarly, East/West Germany were not officially called East/West Germany, and it was not the official name of either country, but people still understood what the words meant.
Not since the early 90s.
Well actually it's a bit more complex since there are multiple parties and they disagree, and there's some nuance (e.g. "one China" but without actually defining what that means, exactly). But "Taiwan claims to be the legitimate government of the whole of China" is woefully outdated and incorrect.