Your question may be based on a misunderstanding of how unified things really were for a large part of the history of China, but let me know if that's not the case.
You need to keep in mind that shortly after being unified, the area broke up again a few hundred years later. The Qin only lasted a couple decades, and while the Han which followed lasted much longer, it was immediately followed by a significant breakup. Later attempts at 'unification' were following different borders, and for the entirety of the history of the area, borders have been constantly changing from one state to the next. China didn't simply unify 200 years before Christ and stay that way to the present.
Really you could argue that Europe was unified in the same time period, or at least he Mediterranean region was, under the Roman Empire. Halfway through the Han, the Roman Empire was covering a larger area than what you're thinking of as China.
While /u/keyilan addressed the fact that the region we know as China wasn't as unified as many people think it was- additionally, the older Empires (Qin, Han) were generally geographically restricted to the North/Central Chinese plains.
Northern China is mostly plains, with a few large east-west navigable rivers. This made communication, travel, and control comparatively easy; the Sichuan basin, Southern China, and other mountainous areas were often harder to control, and broke away to form independent Kingdoms. A good illustration is the Three Kingdoms area- Wei remained the largest, with the manpower and resources of the breadbasket that was North-central China. However, Shu was able to hold out because they were in the Sichuan basin, which is more geographically isolated.
Beyond this, though, China really did tend to fragment and re-unite. Here, the primary difference with Europe may be more of a cultural one. Large European empires had fragmented; Charlemagne created what he called the "Holy Roman Empire", but the idea of continuity, and that All Under Heaven should be ruled by a single Emperor- didn't really exist or have the same currency. The rulers of individual Chinese kingdoms during periods of insurrection generally weren't fighting for independence, or to break away, but for the right to be the successor Dynasty. Even foreign groups who established Empires- the Mongols, Jurchens, Khitans, Manchus- simply ended up becoming Sinicized, and their northern plains became part of All Under Heaven, parts of China. Some groups were more Sinicized than others- this identity and existence as part of the "Sinosphere" gets weaker towards the periphery.
Ironically, it can sort of be seen through a modern lense. The Chinese Civil War kind of echoes previous conflicts- both Taiwan (Republic of China) and the PRC agree on the "One China" principle- there is only one China, that includes all of the mainland and Taiwan (and Mongolia, according to Taiwan). They just disagree about which government has the right to rule it. It's not identical, but a similar principle applied earlier- even when split, Chinese kingdoms agreed there was just one nation, one 天下 (Tianxia), but disagreed who the Emperor should be.
One thing you need to remember is that "Europe" was not a coherent concept until quite recently. The Greek designation of "Europe" was geographic, essentially meaning everything west of the Black Sea, and while there were some cultural implications we shouldn't read much into them. The "European" Romans felt much more at home in the company of Persians than Germans, for example. I would argue that the real origin of "Europe" is with Catholic Christianity, then this got injected with racial ideology of the eighteenth and nineteenth century, was retroactively justified by the wholesale appropriation of classical civilization as a "founding myth" and came out the other side as the cultural identity today. So we don't really have any real sense of Europe as a thing during the Roman period.
An identifier with actual coherence in an ancient context, and arguably modern as well, is "Mediterranean". Its only period of unification was under the Romans, but I can't really comment on why that is the case.
If you asking about the earliest notion of China with the Three Sovereigns and Five Emperors period, the "nation" was still not very cohesive, and multiple tribes and states were noted to have existed during this time, as well as having contact with each other. [Wikipedia has a table with this information.] (http://en.wikipedia.org/wiki/List_of_Neolithic_cultures_of_China)
If you're referring to the general unification of China with Qin Shi Huang: The first thing to note is that while China did have different languages (spoken and written) before and during the Warring States period, where you can find the written systems' names and a [brief summary here if you want to know more] (http://www.chinaculture.org/gb/en_madeinchina/2005-11/11/content_75725_3.htm), by no means did any of the states believe that they weren't "Chinese"(as earlier they were largely part of the Zhou and Shang Dynasties). The idea that they had been a cohesive state still was present in the culture (for example, the jiu ding system that was ultimately fragmented with the Warring States).
The issue with your question is that you are comparing a small area of very similar culture like the Warring States that eventually was expanded to include modern China with an area that had never been united under one banner and was largely separate in cultures. At this point in time, several states existed in Europe:
So in this regard, it boils down to the cultures of the states as they developed. Since Europe had been composed of states that were largely kept separate as they developed, the chance of any one state having enough power and influence to unite the entire continent so early on was low. Without any sense of a unified identity, cultural aversion was definitely a real issue when various states were at war.