Taiwan is not a part of China. Although people in Taiwan are educated "China is a part of Republic of China in Taiwan"
BTW, that is not the issue here.
The issue is the name of China Airlines. It is confusing.
This airlines is not based anywhere in China, and people get confused with Air China.
But I do know that name change is a big deal. All the contract between countries based on Chicago Convention night need to revised. Also, China will not allow this airlines to pass China airfield from Taiwan to Europe or south Asia.
Overall, this is not easy.