So Republic of China means Taiwan? Why??

This is a real conversation took place in Ikea.
We've been visiting Ikea frequently recently. Every now and then W would come up something just to test or to annoy me. I'm not sure if it is what he refers as "to nuture my sense of humour"..

W: Hey, this is made in Taiwan!
A: Huh? That's rare?? (so I closely inspect the bottom of a vase) No, People's Republic of China is China! PRC is not Taiwan!
W: So what is the name again?
A: ( I knew he was mocking but can't help get annoyed by this...) Republic of China.
W: So Republic of China means Taiwan, and People's Rublic of China is China? They both have China in the name but neither of them say anything about Taiwan? Why?
A: ehhhhhhhhhhhh