There is no single perspective on how Taiwanese people view themselves as a culture. Like our population in the United States, there are many different ethnic groups that reside here. Most can trace their ancestry back to ancient Chinese heritage. Others are part of tribes whose histories originated from the island itself. These aboriginal groups have a deep history with Taiwan as their ancestral land. Though they are not commonly represented in the Taiwanese government and media of today, their cities and traditions are as originally Taiwanese as it could get. As for the Chinese populations, their relationship with Taiwan can differ greatly from family to family.
The history of Taiwan is quite interesting, and many have different perspectives to offer. If you look up Taiwan today, you'll find that even their status as a country is in question around the world today. Taiwan's current government is the result of the Chinese Civil War that ended around 1949. The Kuomintang, the name of the government that once controlled all of China, was overthrown and fled to the island of Taiwan where they maintained their claim as the official "China." The People's Republic of China, however, remained on the mainland of China and still claimed to own Taiwan as well.