The logic as I understand it is this:
The long (!) history of China was that when the central government is strong, it expands and controls all the adjacent territory to put it under control of the central government. In times of weak central government, outside forces come in and take part of the territory or conquer the whole, and upstart local warlords hive off their own lands to be independent.
Most recently, the last 2 centuries have made the central government to be weak - being forced to allow westerners to essentially be a drug cartel, taking territories like Hong Kong, Macau and the Shanghai zone - not to mention the Japanese Manchurian encroachment ending in a vicious occupation, and the long civil war.
The current government essentially does not want to appear weak. Hence, its strong response to any territories that were once part of China - it wants to assert sovereignty. Tibet is a good example - at various times it has been part of China, a vassal state, or independent. PRC also does not tolerate independence movements, since they are basically an insult implying the government is weak. By asserting its extended claim to the South China Sea, it presents itself as stronger than all previous Chinese governments.
By this logic, Taiwan - another on again, off again Chinese territory - is implying by its simple existence that the the PRC is not as strong as it should be. Hence the PRC refusal to accept any compromise on the nominal claim that it is a separate country.
As I understand, some Taiwanese politicians want to give up the pretense of being “China” and declare they are what they de facto are, a separate country. However, making this formal declaration would be seen by Beijing as an extreme further insult and deliberate provocation. Why poke the panda - it does have claws.
The most obvious manifestation of PRC’s claim for the average tourist, is that customs will confiscate any materials - maps, travel books - that hint at Taiwan (or Tibet) being separate countries - even if it’s just different colours on a map.