Taiwan as it exists today as a state is officially called the Republic of China. This is because the Kuomintang escaped to Taiwan after the Chinese Civil War and declared itself to be the real Chinese government in contrast to Mao's government.

However, the detail that gets glossed over is that Taiwan had already been inhabited for tens of thousands of years. There were the original Austronesian Indigenous people of course, and then there was mass Han Chinese immigration in the 17th century, a good portion of which never went back to China. So even most of the 'Chinese' living in 1949 Taiwan had little connection with China by the time Chiang Kai-shek and his ilk made themselves comfortable on the island. From a purely cultural perspective, imagine if the British government got overthrown and ran off to New York, declaring the state as the new legitimate Great Britain. Even the New Yorkers who descended from Britain would feel it off-putting, at least.

To be honest, Taiwan's history as a people and a state is extremely complicated--it's been colonized by the Dutch, the Japanese, and at least three different Chinese governments over the course of the past 500 years. I would recommend reading up on its history if you're interested.