Why doesn't China claim all of Japan, for Japan at one point in history conquered most of China?
With that logic, why shouldn't Poland claim France because at one time Poland was conquered by Germany which also conquered France. And Poland can play the other end. Why shouldn't Poland claim all of Russia all the way to Siberia because at one time Russia conquered Poland.
Well, speaking of influence, America can prove that many dictators of other countries have studied in American Universities. So why doesn't America claim those countries?
Furthermore, for China to claim that it has records Okinawa, (formerly the kingdom of Ryukyu), formerly sent tribute to China's Ming Dynasty, does not give China a claim on the land. China receiving "tribute" was actually often the only way other nations could officially trade with China during different periods in China's history. The other nation would bring "tribute" and then the Emperor would in turn give "gifts" to that nation's envoys. As Charles C. Mann ably explains in his book 1493, this was what constituted for trade with China.
Certainly, at certain times in history, Chinese emperors defeated kings of other nations in battles and exacted tribute for a time, but even in these cases, China did not seek to annex these nations, but decided it was easier simply to exact some kind of tribute. Such history does not give present-day China any right to claim these territories.
Article detailing some of the history of the Senkakus
With regard to Taiwan: The RESOUNDING undeniable conclusion is that just because the Manchurian Empire maintained marginal control over western Taiwan at one point in Taiwan's history does not give the current government of China any claim whatsoever on Taiwan.