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The history of Taiwan is quite a bit more complex than that, but the PRC (current government in mainland China) has never controlled Taiwan - it was never theirs.

Taiwan was a Japanese colony from 1894 until 1945 when Japan was forced to hand it over to the ROC (the successor government to the Qing dynasty, which was the last time you could argue China controlled the island - the Qing managed to almost fully colonize it before losing it to the Japanese, although a lot of the mountainous parts of Taiwan were still mostly autonomous at that time and inhabited by aboriginal Taiwanese who continued to resist the Qing rule)

The ROC takeover of the island is also seen as another colonization by many Taiwanese as well - the descendants of the Qing era colonists who were mostly Hokkien speakers from Fujian, while the ROC migration in 1949 was mostly Mandarin speakers from wider China, who fairly brutally imposed their rule over the island (see 4 decades of martial law, etc.)

ROC managed to reform itself over time, and Taiwan is now a vibrant democratic country which is forging its new national identity where most people would prefer to be left alone to control their own affairs.