This is the classic conception: there is the single country that is China, with two regions (the mainland region and Taiwan region) and two governments on each region, each asserting their own rights as the sole and legitimate government of all China. The government on Taiwan lost this competition years ago, and now it has only 13 countries left in the world that recognize their legitimacy, one of which is the Vatican.
However, recent trends in Taiwan is to promote independence on the down-low, to change their mandatory history education to say Taiwan never had anything to do with China, not even historically, that it essentially sprang into existence some one hundred years ago, owes more thanks to Japan for its culture, and whatever cultural elements that are similar to the mainland are just because the mainland had been a large general influence all over Asia.
The DPP doesn’t dare to change the Constitution overtly to make Taiwan an independent country, though they keep treading the red line in their actions, because they know that the day they do so will be the day the CPC is forced into military action. The US also would not approve, as it wants Taiwan to be a manageable thorn in the CPC’s side, not an out-of-control element that will pull the US into a war it isn’t yet ready for.
Arguably, this is part of cultural DNA of the united states, which forebears arrived to escape authoritarian persecution, and whose Constitution assumes the state is there to deprive the rights from citizens, which must be protected from the state via laws, rather than believing that a state can also work to specifically enact and protect those rights, that without a state there ARE no rights.