The problem with using "China" here is that it is always misleading if people have no prior knowledge of the situation.
Specifically "China" here means "People's Republic of China" because they consider that the "Republic of China" ceased to exist in 1949 and Taiwan, as a province of the Republic of China automatically moved under the PRC sovereignty. Conversely the Republic of China (in Taiwan now) does not officially recognises the PRC.
So it's a bit like if the two Koreas refused to acknowledge the existence of each other and claimed the whole peninsula for themselves, which I believe is actually not far from the situation there. But it's more complicated with China because of the existence of a "3rd faction" in Taiwan that would like to see Taiwan independent of any Chinese state whatever that state might be.