There are many factors that play into how Chinese and American people see their identities. Chinese people pride themselves on being a civilization with 5,000 years of history, and hence everyone who is born into a Chinese family, no matter the geographic location, is obliged to always identify him or herself as Chinese. Otherwise it’s a dishonor to the family and what have you.
On the other hand, other than the Native Americans, the U.S. is a country of immigrants, who left their home countries in hope of finding a better life in the states. They may retain the cultural heritage of their country of origin, but they and their descendants primarily identify themselves as Americans (e.g. Asian American, African American, European American, etc.).
We then come to Taiwan, an island that was only inhabited by indigenous Taiwanese peoples for thousands of years before Chinese immigration started. The influx of Han people for about 500 years make up Taiwan’s population today, and we see the conflict between the two attitudes above. On the one side, there is the attitude of one must always remain Chinese because of the Chinese ancestry (this is especially relevant among the group that immigrated in 1949). On the other side, there is a strong affinity with the island, and we see that in people increasingly identifying themselves as Taiwanese.
Most people in Taiwan today can probably be called Chinese Taiwanese (analogous to Chinese American in the U.S.), but as the KMT regime after 1945 actively suppressed a Taiwanese identity, that label didn’t really catch on. Instead, we now have Hoklo Taiwanese, Hakka Taiwanese, Indigenous Taiwanese, etc. What is absurd is that a small percentage of people who only wish to identify themselves as Chinese would like the rest of the population to only identify themselves as Chinese, despite a large proportion of them not minding identifying as culturally Chinese in addition to having Taiwanese nationality. When people are forced to choose between only identifying as Chinese or Taiwanese, it is perhaps not that surprising that ultimately the majority of people would choose Taiwanese over Chinese.