The same thing happens for North and South Korean airlines, they do not fly into each others' airspace because their governments do not recognise each others' legitimacy or sovereignty. This is of no importance to North Korean airlines (oops, airline) because they have no need to overfly the South, but for Southern airlines they always stay well clear of the border when taking off or landing at ICN.

Btw, neither China or Taiwan officially consider Taiwan to be a separate country. Both governments claim they are the legitimate government of the whole of China and Taiwan and do not recognise each other. So in this sense, Taiwan doesn't exist as a country de jure. Of course de facto it is.