The "Taiwan is part of China" has been in the text book of China since the People's Republic of China founded. If you looked at diplomacy, one thing China did previously was to maintain the status quo to a degree that it has refused to take the few countries which recognize Taiwan in order to maintain the friendly relationship.
I have to say that now things changed. That's why it has become a hotspot for a potential war zone between US and China.
As for TSMC, I believe the decision should be attributed to Biden's government's push for having production capacity in the US rather than a pure business decision.
Well... The island has been a part of China historically, taken by the Japanese in the first Sino-Japanese war, and after WWII returned to China, which was in the middle of a civil war, in which the losing side ran away to the island. Their flag and official name are the ones of the Kuomintang party and regime, aforementioned losing side.
It'd be like if the CSA run away to Hawaii, assimilated the local population, called itself CSA with flag and all, and the USA still said from times to times it belongs to them.
And not only that, the UN and all members see it like that as well - officially and for diplomatic purposes, Taiwan doesn't exist and is a part of China.
They badly want Taiwan. Not at any cost, but certainly at the cost of hundreds of thousands of lives —- especially those of the ‘ungrateful’ and ‘unruly’ Taiwanese.
The existence of trade between them in no way contradicts those desires.