It's possible, just as it would be for a European country, for instance. China is a nation, in English and in French, but it also refers to a culture, and the Taiwanese openly admit that. Even though the most recent figures show that 64% of them identify as Taiwanese whereas only 20% identify as both Chinese and Taiwanese, the Chinese identity still matters. It represents their culture and the language they speak. Fukienese, the language spoken by the majority of Taiwanese, is, after all, the language spoken in the nearby province of Fujian.
The Taiwanese are willing to admit that, culturally, they are Chinese, so it may not be as preposterous as it seems for Taiwan to be named the Republic of China. It does have a basis.