The Republic of China is in control of the island formally known as Formosa. Taipei is a city on Formosa also under Republic of China control. Internationally, representatives of Taipei and the surrounding area are sometimes said to be representing "Chinese Taipei", because Taipei is on Formosa and Formosa is Chinese. There is only one China, and China is the rightful owner of Formosa, and one of its cities, Taipei.
There are also plenty of Taiwanese that don’t think Taiwan belongs to either the PRC or ROC. The DPP, who are in power right now, is much closer to that opinion. Obviously the CPC dislikes the DPP (compared to the KMT, who at least believe Taiwan is a part of a china), but whenever the CPC speaks out against the DPP, the CPC only helps the DPP win elections.
China just uses its weight to bully countries into submission on declaring their view on Taiwan.