Taiwan continues to keep the name because it mustn’t give China a cassus belli to invade. Essentially, by keeping their name, they show their support for a “one china policy” and keep the door open to peaceful reunification. If Taiwan decided to formally be a separate thing, the PRC would no longer be able to “peacefully reunify with a rebellious province,” and would be much likelier to invade.
https://www.reuters.com/article/us-taiwan-passport-idUSKBN25T0JA/
(I don’t care for Reuters, but the facts are easily verifiable elsewhere)
The PRC got pissy when Taiwan changed its passports to read “TAIWAN” in big letters and have “Republic of China” in small letters. What purpose would they have in being mad about a name change on a foreign nation’s passport (particularly as it de-emphasizes the “China” bit)?
Furthermore, if you look into history, Taiwan got rid of its claims to “all of China” decades ago
https://www.taipeitimes.com/News/taiwan/archives/2002/09/10/167505
Premise reminds me of The Integral Trees a lot