There are a lot of people on the island that were there before the KMT arrived that don't consider it China or themselves Chinese.
The thing to realise is that this is really a divide and conquer game by the West.
The West has always recognised that Taiwan is a part of China. Not least of course when the recognised government of China was the ROC government.
Now the West recognises the PRC as the government of China, but this means that trying to stroke an 'independentist' sentiment in Taiwan becomes a tactic to weaken them.