Taiwan is the common name for the state that officially goes by the name Republic of China. It's de-facto administrative boundaries have been the island of Taiwan for the last 70 years. That's long enough for most people stop caring all that much. Many countries in todays world were still colonies back then.
The ROC had existed continuously since over 30 years before it occupied the island of Taiwan. What about people in Kinmen? Do you consider them part of "Taiwan"? Please use the official government name to stop aiding the PRC in its misinformation campaign.