I'm fairly certain the people of Taiwan consider Taiwan a different place than China -- enough so that they have the whole Taiwan name and all.
Actually It's a moving target over 40 years or so...
From the end of WWII the occupiers of Taiwan (basically the retreating/invading Chiang Kai-shek govt) pretty much considered themselves the exiled government of mainland china, thus calling themselves the Republic of China. The ground started significantly changing in 1971 when after UN resolution 2758 passed, mainland china (aka the People's RoC) was able to reclaim their UN seat which. Eventually, the notion that the RoC (aka Taiwan) was a different place than china all but faded by 1991as by then most in the RoC conceded that mainland china was lost (to the PRoC) by forcing the resignation of the so-called "representatives" tied to legacy captured provinces in the mainland.
Of course as with most things Taiwanese, it ain't that simple.
Some of people of Taiwan were repressed by the retreating CKS occupiers from the mainland (not much different than the Japanese), but that distinction is often not understood by those outside of Taiwan (see the 228 incident). If you know people from the south part of the island (e.g, Kaoshung, or Tainan), many still hold a grudge, and think of both remnants of the CKS government and the mainland as the enemy. These folks form the basis of the pan-green coalition (not to be confused with the environmental green party movement, but one favoring Taiwan independence) to oppose the pan-blue coalition (remnants of the CKS/KMT government + other parties favoring close ties with the mainland).
Of course, the people of mainland china don't see things that way at all. The see it as formosa island/taiwan provence which was historically part of mainland china (except for the time the Dutch and Japanese occupied it, of course). RoC is generally pissed about how the US handled the disposition of Taiwan after WWII (with the Treaty of Peace with Japan aka Treaty of San Francisco to which the RoC and PRoC were not invited). Basically the island of Formosa/Taiwan was treated the same as occupied territory whose responsibility was given to the US, but whose final fate was undecided (much less complicated, but similar to Berlin). In contrast, the Treaty of Taipei (a separate peace treaty between PRoC and Japan) further complicated the matter by obfuscating the issue of Taiwan by reclaiming it for the PRoC even though Japan had no authority to grant it at that point having ceded authority over Taiwan in Treaty of San Francisco...