Having spent some time in Taiwan, the people there refer to themselves as Chinese, not Taiwanese. They don't speak Mandarin, they speak Chinese. Which is to say, their national identity is "Chinese" and that's what works for them. They also have a standing military that is completely independent of PRC.
That said, the only reason they can maintain that independence is because of how close they are tied to the US. RoC knows it, we know it, and Mainland China knows it.