It has to do with the idea of democracy (or in Rome's case, republic), and the rights given to those who could claim citizenship. Roman and Greek citizen enjoyed numerous rights that non-citizens did not.

Meanwhile, China never had that kind of philosophical underpinning. For virtually the totality of its 5000 year history it was a feudal monarchy where absolute power was derived from heaven itself. In fact, one of the titles conferred onto the reigning monarch was 天子, literally the "Son of Heaven." There was no conceptualization of individual right or liberty, as unlanded peasant and top-tier lord were equally subject to the whims of the absolute ruler of his or her day, along with all additional layers of bureaucratic authority against which there was little to no recourse. Often times the very act of being accused of a crime was enough to warrant harshly punitive action, or drive those to rebel in fear of it... especially during the more legalistic Dynasties such as Qin and early Han.

That said, there was a large emphasis on being Chinese, and an acute awareness of foreign-ness. The most common names given to those who lived on the peripheries of the Empire's borders were 野 (Yě) and 夷 (Yí), both most commonly (and accurately) translated as "barbarian." Notably, though, this was not based on our modern ideation of race. Sure, there was a bloodline component of being "Han/Huaxia" ... but typically that was some silliness tracing one's lineage back to the fictional Yellow Emperor. Rather, whether or not you were considered Chinese (much as with the "Roman-ness" of the Western Empire's citizenry as its borders grew ever outward) was if you looked and behaved like a Chinese. If you did, for enough successive generations, congrats you were in the club. If you hung onto your individual sovereign traditions, then you'd remain stuck in cultural limbo, branded semi-barbarian, and frequently denied higher honors or promotions from the Imperial Throne (this is particularly notable, for instance, during the Warring States Period when the Kingdom of Chu sought to gain more prestige, it's Chancellor Wu Qi enacted building reforms make its capital come more into line with Chinese architecture so it wouldn't be seen as a barbarian city.)

To the north, the proto-Mongolians were called by the Chinese 匈奴, or "Xiongnu." This, as with other components of their culture that are filtered down to use through the completely biased Chinese sources (since the Xiongnu were pre-literate, they have no records of their own), is was likely initially derived from what they called themselves, but was soon thereafter homophone-swapped to the far more derisive "fearsome slave" ... a name that would stick even after the two cultures came to military equality with each other in the 1st century BCE through 1st century CE.

So, circling back around from that substantial detour I've just taken, yeah, the idea of citizenship was kind of a big deal for Chinese... but not in the same way or for the same reasons as Rome/Greece. This is because there were no additional rights conveyed to citizens versus anything else. Oh sure, it meant you weren't a slave, but the top-down flow of political power meant that any position you held at any level was given to you by the grace of the Emperor's will, and there had never emerged the necessary competing political philosophy to challenge that idea, as had arisen in the Peloponnesian peninsula, and later the city-state on the Palatine Hill.

P.S. If China's a subject that interests you, consider checking out The History of China Podcast.

Citizenship, and the "big deal" about it in Europe's antiquity was dependent upon the possession of it as a marker of elite status (as opposed to merely being a freeman or a slave) and the rights to its particular privileges, such as the defined legal right to partake in the political process.

Whereas in Qin and Han China, though they also had elite statuses and ranks with privileges, there was no legally defined privilege of participation in the political process. That doesn't mean that elites had no say in the political process (as exemplified by the many accounts of factional jockeying over access to the emperor and the selection his successors), it just means it wasn't legally defined.

Which one could argue, is part of a philosophical difference between the west and the east, namely what should be the more significant part of what is prescribed by law: rights vs. obligations.

Reinforcing the point about citizenship's "big deal" being dependent upon it's elite status, when Caracalla issued his edict giving citizenship to everyone in the empire (to boost tax collection) it quickly ceased becoming a primary marker of personal identity, as it had instead became a universal obligation instead of privilege.

I can't comment on China, but I'll say this on citizenship in Greece and Rome:

Regarding the Greeks, they did highly treasure citizenship and protected it to their utmost. There really was no way for a non-citizen to become a citizen, you had to be born into it from at least one citizen parent (and both parents in Periclean Athens). I would correct /u/cthulhushrugged that this xenophobia was not due to democracy as it was common with all Greek poleis - democratic or oligarchical.

Regarding the Romans, citizenship was much more open. Rome's success, in part, was built on their ability to integrate conquered peoples into their empire. Elites would be wooed by wine, access to trade, sophistication like baths, bridges, aqueducts, etc... This would sway them to adopting Roman culture and lifestyle - and that's what mattered to Rome.

Generally speaking, Rome didn't care about your race or state but whether you adopted Roman cultural values and (under the empire) participated in the Imperial cult. Citizenship was often handed out as rewards for provincial elites which brought significant advantages.

The unique thing was that this openness with citizenship also occurred at the bottom of the hierarchy - slaves freed by a Roman master automatically became citizens (except they could not run for office some political offices). Their children had full citizen rights, including the ability to run for any office (though it should be noted that their success rate for something like the consulship is equivalent to my chances of dating Scarlett Johansson). And the Romans had a habit of freeing their domestic slaves (but agricultural slaves on large 'corporate' farms or slaves working in a mine were usually shit out of luck) - so much so that Augustus put a tax on manumissions to try to slow down the rate.