This idea that "the Mongols assimilated into Chinese culture" is a modern idea and concept, though at what point it was conceived I'm not sure. It's not that there isn't some truth to this idea (but I'll get to that) but it seems to me far more rooted around squaring the root of the modern (Chinese) Republican idea of the Five Races with (practical if not explicit) Han Superiority. In a nutshell, the Mongols are one of the "Five Races" therefore the gifts of Chinese civilization naturally flower outward towards them.

At the time and on the ground, however, that's not how Qhubilai Khaan would have seen things, nor how the Mongols would have described their own process. A quote from George R.R. Martin's A Dance with Dragons I think is quite relevant here: "If you want to be king of the rabbits, you have to wear floppy ears." In other posts (see my recent one about the literary form and archetypes that Mulan has played over the centuries) I have written about how Buddhism was adopted to form imperial legitimacy on "barbarian" states that conquered China, beginning with the Tuoba, then the Mongols, and then the Manchu. But any power that conquered China still had to contend with deeply held and understood notions and beliefs regarding Confucianism and Taoism. When native Chinese dynasties were in control, they upheld those traditions first and foremost, and Buddhism was placed back among its "equals."

The Mongols, throughout their reign, actively kept themselves and their traditions separate from the Chinese. For example, Mongol language was illegal to teach to outsiders. Many territories that were explicitly Mongol were forbidden to Chinese, in particular. Quite literally, the place we now call "the Forbidden City" began as a Mongol construction of walls that could function as the Steppe in miniature. The Borjigin kept the area open and moved their gers around the relatively small enclosure as they would out in Mongolia. The Mongols didn't adopt Chinese script, and actually commissioned a Tibetan Lama, Drogon Chogyal Phagpa, to develop a script that would be legible and translatable between Tibetan, Mongol, and Chinese (this complex script was abandoned after the fall of the Mongol Khanate, but was studied by Korean scholars who later developed the Hunminjeongum, the predecessor to the modern Hangul script). Even when the Mongols did adopt Chinese for their own purposes, it was incomprehensible to the Chinese. Literally. The rather famous Secret History of the Mongols was "secret" because it was written in Chinese, but only understandable if one spoke Mongolian (i.e. if one read out the Chinese characters with which the Secret History was written, it would make little sense in the Chinese, but to a Mongol ear would be perfectly comprehensible). The Mongols also developed their own script for their own purposes, which survives today in archaic Manchu stuff (still visible on Chinese Yuan bills, along with Chinese, Tibetan, Uyghur, and Modern Mongolian written in Cyrillic script), but was used by the Mongols until the 1940s when the Soviets introduced use of the script and sought to tie the Mongolian People's Republic economically and culturall to Russia.

Once the Ming took over in 1368, they began the Sinification of the territory controlled by the Yuan as best they could. The Hongwu Emperor, formerly a Buddhist monk who lived in abject poverty, was known for his cutting of state expenses, and frugal living (at least by imperial standards). The Ming actively Sinified things around them, for example, the Forbidden City was no longer a mini-Mongol Steppe, but Chinese-style palaces were constructed into the vast complex. Where the Mongols had patronized or themselves studied Chinese culture, like in their works on the Chinese Classic Yijing, the Ming actively suppressed them and favored works that were developed in the Song, Tang, and Han Dynasties. (When the Manchu took over in 1644, they actively pursued a connection with the Mongol Borjigin Dynasty of the past and would revive the Yuan-era works on the Yijing during their own long, and diverse period of Yijing-scholarship, while more-or-less ignoring or at least putting on an equal footing, those works composed during the Ming era.)

According to, ahem, Jack Weatherford, there appears to be at least a modicum of truth into the "assimilation backfiring" though this to me seems a tale amplified by Western understanding of Chinese history. The modicum of truth, as far as I understand it, however, is that when Temujin's sons began taking over China, they came into contact with Chinese wine. Normally, good Mongol men drank fermented mares milk. This drink Mongol temperaments and stomachs had become accustomed to. One cannot drink so much of it before drowsiness sets in. Wine, however, can be drunk much to excess, and the nomadic Mongols had little tolerance for the stronger and sweeter brew. Weatherford is rather critical (and imo, rightly so) of Temujin's sons who were "good for fighting and little else." And this difference in the potency and availability of alcohol in indigenous nomadic cuisine vs. those available in the conquest of China is a vast oversimplification, but is at least emblematic of the challenges in aligning the two concepts.

Western contact with China often had a lot to say about the decadence of Asian governments (without a hint of irony, obviously) and China being the most magnificent of all of them, they had a lot to say about the imperial throne. When it came to the Mongols, there was often a struggle to understand how the fierce and bloodthirsty barbarians could fall to such a fat and wasteful people, and the concept of the decadence and assimilation of the Chinese cultural sphere was brought into the conversation. The Republic of China which rose in 1911 was already inundated with evolving Western ideas of ethnicity, nationalism, and republican sentiment, and one of the first actions that signaled the rebellion was mass cutting off of the queue. The queue, which you'll recognize from any image of China or the Chinese in the 1900s, was a length of braided hair at the back of any Chinese man's head. The queue was not a style choice, but a legal requirement imposed by the dominant Manchu ethnicity and class above their Chinese subjects. By cutting it off, a Chinese man was committing a crime, and when the revolution began, it was often with the aggressive cutting off of the queue. Soon afterwards, this Han Nationalism evolved to lay the groundwork for why the Republic of China should inherit all of the former territory of the former Manchu Qing Empire, even though far more than half of that land was inhabited by nations undergoing their own nationalism processes: Tibetans, Turks, and Mongols. Thus, the concept of the Five Races (Han, Manchu, Mongol, Tibetan, and Uyghur) united under "one China" was born, and the decadence of the assimilation into Chinese culture used to explain why the Mongols ultimately fell.