It stems back to Kaishek. When China and Taiwan WERE together. It is you who seems to be confused. He kept the name ROC because he planned to take back all of China from the communists. Not just the PRC, but also the ROC, agree.
International treaties agree.
The territory of the Republic of China according to its existing national boundaries shall not be altered except by resolution of the National Assembly. Those same treaties that, much like Neville Chamberlain and Hitler bargained away Czechloslovokia without the consent of the Czechs. That s what happened in 1979. The US withdrew formal recognition of Taiwan, which it recognized as independent for nearly 30 years, in favor of China to appease the Chinese government to secure trade relations. The US had rather recognised the Republic of China, currently in headquartered in the remaining free are of the Republic of China, i.e. Taiwan, as the legitimate government of China.