both view themselves as the one rightful Chinese government

This is a bit of an outdated view in my opinion. If you're a KMT voter (ie. 60+), then sure, this is a common view. Younger generations (DPP voters) however don't really view themselves as Chinese. I think this view will die out eventually.

Of course, for the most part, this is all off the official record because of the implications. Chinese nationalists will argue that this is wrong because it's still written in the Taiwanese constitution or whatever, but the truth is that regular people in Taiwan couldn't give less of a shit about China. De facto, most Taiwanese consider themselves their own country with no legitimate claims to China.