That's the entire point I think: Chinese cultural identity is broad enough to consider all those numerous competing kingdoms of its past predecessors in direct line. Contrast this with Europe, where almost everybody has at some point in time claimed succession to the Roman Empire but even something as small as a united Italy is actually a fairly recent invention, with forced language unification and all that.
Egypt, to take an example GP used, had numerous cultures and languages sweep through its geographic location so that modern Egypt has nothing in common with the one 5000 years ago except for the fact that some pyramids are still standing.
Perhaps the secret to perceived Chinese contiuity is (besides considerable geographic isolation) the non-phonetic writing system which should be relatively stable when dialects inevitably diverge when regions shift apart politically for more than a few decades.
I don't think that the Taiwan situation can count as a counterexample, actually I'd rather see it as a contemporary manifestation of "the idea of China is bigger than its governments". The way I see it the main issue preventing a normalization of relationships between continental and Taiwan is that they agree on one thing that is important to them: they both are far too Chinese to be anything but China. The fact that there is more than one operational government within China? Not that big of an issue, happened before, will likely happen again.
The Roman Empire was an overarching idea whenever it was fractured, but to Kaisers, Czars, Sultans, that one guy from Corsica and whoever else claimed spiritual succession it was just a claim to status without a trace of cultural continuity.
Rome has been a hugely important influence to many cultures and nations, but the identity fanned out into numerous branches to thin to bear identity:
The city itself, well, that's still standing but it's certainly no empire.
The (western) administrative network, in a freak turn of events that you couldn't make up if you tried, lives on in the Catholic church, headed by that one office of republican Rome that kind of still exists. Pontifex Maximus, head priest in charge of making up calendar rules to strike fear upon software developers.
The political entity, it lived on far into the middle ages. But how Roman can you be when you don't talk a word of Latin?
The various populations who did talk Latin: power positions taken over by foreign warlords with little talent for or interest in administrative continuity (which is how the church got to monopolize all clerical applications) and by the time they had assimilated with the locals, cultural identity had long transformed into something new.
The Frankish peregrine kings and their successors who occasionally got an outmaneuvered pope and the acclaim of bribed and/or terrified townspeople to declare them emperor? Not much Roman about them, not even much territorial overlap. The continued existence of the actual Roman Empire somewhere else did help either and those two certainly did not agree that they were one.
The brand that we think of when we hear "Roman Empire" is Cesar, Cicero, their language and their gang of pagan gods. But that's just a snapshot, it ignores centuries of Christian empire, the many centuries of Christian Greek empire. The cultural identity died before the political entity, perhaps this is pay of why the echoes of the cultural identity are so strong.
Egypt, to take an example GP used, had numerous cultures and languages sweep through its geographic location so that modern Egypt has nothing in common with the one 5000 years ago except for the fact that some pyramids are still standing.
Perhaps the secret to perceived Chinese contiuity is (besides considerable geographic isolation) the non-phonetic writing system which should be relatively stable when dialects inevitably diverge when regions shift apart politically for more than a few decades.