The Republic of China’s government (which controls the island Taiwan) officially claims to be the legitimate government of all of China. The mainland is a rebel province from their point of view. Until 2002, their definition of “all of China” included Mongolia, and they included it in their territory on official maps. By comparison, the People’s Republic of China government (which controls mainland China) recognised Mongolia and established official diplomatic relations in 1949. Until recently, the RoC officially claimed more territory than the PRC.

That’s my point: there is no country called “Taiwan”, it’s the name of an island. There are two governments that claim to be the legitimate government of all of China, one of which (the RoC) controls Taiwan and a few other islands, and the other of which (the PRC) controls the rest of the current territorial extent.

Now a significant proportion of the population of the island Taiwan would prefer if the island Taiwan was recognised as a separate country. I’d guess the majority of Taiwanese residents under 40 feel that way.

There are two major political parties in Taiwan: the Kuomintang (KMT), or “blue party”, and the Democratic Progressive Party (DPP), or “green party”. The KMT leans heavily towards the view that there is only one “China” and we’re just still arguing over who’s the legitimate government, while the DPP leans more towards the view that the island of Taiwan will eventually be recognised as an independent sovereign state. The current president of the RoC, Tsai Ing-wen (蔡英文), has said unofficially that Taiwan is an independent country and the PRC has to deal with this. However, making this official would require rewriting the RoC’s constitution.