Each successive dynasty(s) of China has claimed its legitimacy by being a successor of the previous state, while retaining a "Chinese" identity. For instance, the Yuan (Mongols) adopted Chinese institutions and ideas in order to legitimize their rule over China. The "Mandate of Heaven" was what an emperor claimed after deposing the previous dynasty. The idea behind the Mandate was that if a ruler became incompetent or failed his duties, the people should revolt and overthrow him. A ruler that managed to repacify China would then clearly have the mandate of Heaven(or God) to rule, thus becoming the legitimate successor to the rule of China. It was first used by the Zhou dynasty as a way to claim legitimacy from destroying the Shang dynasty-namely, "Because China was fracturing under the previous emperor, and now we are united and at peace again, this is Heaven's way of approving my rule. Do you dare to disagree with Heaven?" Similarly, Qin Shi Huang claimed this as well after subjugating the other warring states.

I would contrast this to the Ottoman conquest of the Byzantine Empire. Although the Ottomans claimed to be the successors of the Byzantine Empire (by occupying their former territory), they did not adopt Byzantine institutions or Greek ideas, nor had outright conquest ever been a legitimate method of becoming emperor (or Basileus) of the Byzantines/Rome.

This is, in its way, a highly political question, as you might imagine and there are many different ways to approach it. Over the last 2000 some odd years, the boundaries of 'China' have changed, generally grown since the time of the first Qin emperor, but between the individual dynasties areas of control have grown and shrunk, ruling houses have come different areas and with different ethnic and linguistic backgrounds and while there there is long history of relatively stable dynastic rule, some of the periods of disunity were really pretty long. To someone coming from the perspective of European history, it's totally understandable that it may seem strange that different governments over different areas with different rulers may all be grouped together like that. Partly I suspect that, in the west, there's a certain laxity in the way that people speak about non-western history that comes about as the result of a lack of familiarity. I see casual sources talk about the history of India in similar ways, seeming to group disparate historical entities that came with very different cultural and political baggage under the one heading of 'India'. There's another aspect to the way that Chinese thought has treated 'statehood', though, that I think is much more interesting to consider.

The idea of the 'nation-state', an entity with a clear, inviolable geographic place along ethnic, linguistic or cultural lines, is actually very modern, even in the west. Italy and Germany, as we know them, for example, are products of the modern era. They had not previously been unified politically speaking (not unless you go back to antiquity) and the borders between the smaller kingdoms and principalities had not been set. In China too, until relatively recently, there wasn't really the modern notion of a 'country' or 'state' as independent of the regime. When Jesuits and other western observers started studying and talking about China during the Ming dynasty, this was one of the things they noticed actually. The Chinese didn't really have a name for their 'country'. If asked where they were from, they would say they were a subject of the Great Ming, or the Great Qing, or whatever dynasty.

The historical and political perspective wasn't state-centric in China, rather it was universal in vision. The operative term in Chinese thought was tianxia, 'all under heaven'. In practical terms, there were limits to any given dynasty's ability to project its power, but ideologically and philosophically, the emperor ruled over existence itself. This idea ran throughout all of China's interactions with other nations. The emperor did not recognize other rulers as being of equal sovereign status; the emperor was the 'son of heaven' and others cam to bear tribute. See, for instance, the opening of the Qianlong emperor's response to the Macartney mission: "You, O King, from afar have yearned after the blessings of our civilization, and in your eagerness to come into touch with our converting influence have sent an Embassy across the sea bearing a memorial [memorandum]. I have already taken note of your respectful spirit of submission...." http://www.history.ucsb.edu/faculty/marcuse/classes/2c/texts/1792QianlongLetterGeorgeIII.htm Did King George realize the mission would be taken as a sign of submission?

It was easy for Chinese historiography to treat all the different dynasties with their different borders, ruling houses and political regimes as one subject. This wasn't the history of a country, after all, but the history of the world. Chinese conquests were often described as 'returning' somesuch location to it's rightful place in the empire, or other similar terms. Chinese histories, therefore, don't treat Qin, Han and Tang rule over a given place as being distinct in the way that a western history would treat, say, Roman rule and rule under Charlegmane as being being distinct, even if there is geographic overlap.

There's a lot of reasons for this, including the continuation of founding philosophies throughout the entire stretch of Chinese empires, the continued acceptance of the office of emperor, the presumption of legitimacy of unitary rule of the civilized world, the confucian bureaucracy, the continued use of the chinese script, etc.

However I would like to make an additional argument, based upon the difference between what we westerners call China, vs. what the Chinese call themselves.

"China" is a foreign word, with various guesses as to its origin, the most common one being derived from Qin Shi Huang.

However, the Chinese themselves call China "ZhongGuo" which literally means "Middle/Central Kingdom/Country/State(s)." The amorphousness of this designation means "Central Kingdom" can be viewed geographically instead of nationally, although this itself might be a bit of a tautology from the ancient Chinese perspective because the "inhabitants" (called ZhongHua) of the "Central Kingdom" are naturally "Central Kingdom-ese."

Combine this with the unmooring of the Chinese script from pronounciations, means that unless a brand new conqueror of the land decided to dispense with the cultural, philosophical, bureaucratic, and linguistic apparatus in China all at once (which none of them did), they would be committed to using the terminology of the region.

And since "Central Kingdom" is not quite as ideological, it was never dispensed with.