How can the PRC claim to own ANY portion of Taiwan/Formosa? Is it a lie?

If you want to use history to prove your point, then you should go back far more than 1870's. Before Mao defeated Jiang Jieshi, who fled to Taiwan, Taiwan had never been claimed as an independent country. Going back a bit further, before China was invaded by united army of foreign countries, Taiwan had never been not part of China. Going back even further to Ming Dynasty and before, Taiwan was always ruled by central government of China in every aspect. Whether Taiwan is an independent country or not now is a matter between Chinese people, between mainland China and Taiwan. It had nothing to do with Japense and will have nothing to do with Japense ever!!