The government that administers Taiwan calls itself the Republic of China. Leaders of the government moved to the island from the Chinese mainland in 1949, when Communist armies gained control of the mainland and established the People’s Republic of China (PRC). The government on Taiwan recognizes the mainland city of Nanjing (spelled Nanking in Taiwan) as its official capital, and designates Taiwan’s largest city of Taipei as its temporary capital. The PRC does not recognize the government on Taiwan and considers the island a renegade province. Taiwan recognizes that the Communist government rules the Chinese mainland while the republican government rules Taiwan.

It is not as simple as saying that the “government in Taiwan” is a separate government from that of China mainland.

You must understand that the government of the People’s Republic of China began on Oct. 1, 1949.

Then, for an overview of the history of Taiwan, see — http://www.taiwankey.net/dc/tmodhiae.htm

In very simple terms, at the end of WWII, Taiwan was in a state of disorder after the defeat of the Japanese, who had ruled over it since 1895. When Chiang Kai-shek was beaten by Mao Tse-tung for control of China in 1949, he and about 2 million refugees fled to Taiwan, where he set up a nationalist government supported by the Americans (who were fighting against the spread of communism.)