What happens is both sides have special procedures to deal with each others which allow to control and enable travel and trade without recognising the other side's claims while recognising people as Chinese, not foreigners.
Tricky subject here. The statement might sound true but has very very different sentiments on both sides.
For the Chinese gov., Taiwan is a province of China, "the rightful Chinese government" extends its jurisdiction over all Chinese territories, including Taiwan. However, from Taiwan's perspective, Taiwan and mainland China have two different governments (the stress is on "Taiwan is not part of China", not there is only one rightful Chinese gov,). In fact, if you ask anyone in Taiwan today, they are more likely to refer to the island as Taiwan (instead of "Republic of China") and themselves Taiwanese.
When dealing with legal matters (like travelling or trading) this is what counts.