It would be nice if you could go into more detail about how different dynasties which ruled China were similar and different, and how the basic notion of the Chinese state has stayed pretty much the same over time. I understand that your focus is not on the current physical boundaries of PRC, and I am glad that we are not taking the discussion in that direction.
I'm not sure I could do the topic justice... not sure I'm qualified for such a broad summary. I'll do my best, just go easy on me if I muddle through this.
In the near-mythical past, the Chinese trace our history back to a number of smaller tribes/kingdoms in what is now central China. But in actual recorded history, we have to look to Qin Shihuang at around 220 BC... he's really the father of the Chinese imperial system. He effectively unified all of China, and set into place intellectual traditions that have defined every dynasty since. He decided that he had surpassed the "huang" and "di" kings of the past, and therefore declared himself the first "huangdi" (emperor). Huangdi have the "mandate from heaven", the pseudo-religious right to do anything they wish as absolute dictator. A very complicated bureaucracy of court officials was formed around him. Emperors had absolute administrative power, and appointed officials at every level.
Every dynasty that has followed the Qin Shihuang has taken on the same form, calling themselves huangdi and adopting the exact same structure as the original emperor. For the past 1000 years, there has been a steady trend of "foreign" invasion as nomads invaded the central plains from northern Asia: the Jurchen, the Mongols, and finally the Manchu. It took the Jurchen and the Mongol a couple generations before they were "sinized", but their kings inevitably all declared themselves huangdi, and adopted the same traditional + intellectual court structure that existed before their arrival. The Manchu were even more anxious, they declared themselves huangdi and joined themselves to the imperial tradition almost immediately upon crossing the border. (Perhaps China was fortunate in a way that the Indians weren't... in that none of our immediate neighbors were empires with a lasting tradition/culture like the Muslims or Greeks. To our "barbarian" neighbors, adopting Chinese civilization/culture was very desirable, and the only thing that made sense.)
Note that Qin Shihuang was not a believer in Confucian philosophy; the state 'religion' was instead legalism... which you might even call a cousin to modern facism. A well defined set of firm laws defined the responsibility of every citizen, in service of the state, and anyone who disobeyed would be harshly punished. He attacked and attempted to destroy all opposing philosophies, including Confucianism. After his death, Confucianism quickly surged back to the surface... but the ultimate result was something that's a mixture of legalism and Confucianism. Confucianism essentially states that a man, a family, and a society will thrive if we constantly strive to self-improve, and behave according to certain ethical rules. I would say, legalism resembles something like a top-down, trickle-down view of society; Confucianism is sort of the bottom-up, trickle-up view of the same. Emperors used both to complement each other: obey my laws absolutely, because it's the ethical thing to do.
There is a cultural continuity here, sort of similar to what India experienced. In the case of China, this cultural continuity eventually converted foreign invaders... and that helped maintain the underlying, fundamental political continuity, which is different to what India experienced. Even if a new dynasty came into place... the new imperial court had similar ranks for their officials that the old one had; government had similar "departments"; officials were appointed to rule simliar geographic areas; taxes were collected in the same way; petitions were passed up the same way; documents were written in the same language and structure. And with the Confucian influence here, of course every emperor performed similar rituals. That's not to say details didn't reform along the way over the past 2200 years, but the overall structure remained static. For example, the Qin emperor in 200 BC had six official seals (called xi); after the Sui dynasty in 600 AD, all subsequent emperors had eight official seals... but the these official seals were used in basically the same way.
Ever since 200 BC, China has either had only one emperor (deciding everything), or was in the process of trying to figure out who could claim the title of the next emperor (in order to decide everything). This is clearly very different from the Indian experience. In terms of actual policy, of course every emperor/dynasty was different. Some were militant, some were passive. Some were alcoholics, some were intellectuals. But the system remained in place around them. This makes the events of the 20th century pretty interesting... some, of course, think it represents the end of history, and imperial China has forever disappeared. Others think the Communist Party represents yet another (perhaps short-lived dynasty).