I still am not so sure what the Agatean Empire actually is. The Forbidden City, the Winter and Summer Palaces, the Red Army, and the terracotta golems warriors that Rincewind awakens all suggest China. Of course, there are also Winter and Summer Palaces, as well as a Red Army, in Russia - and the Red Army's plans to storm the Winter Palace are VERY reminiscent of the Bolsheviks - but the fact that they eventually storm the Summer Palace are much more Chinese, since the Summer Palace was actually stormed in 1840, during the Taiping Rebellion. On the other hand, there is talk of ninja stalking the night, the Emperor and his inner circle have a tea ceremony, they use chopsticks and the Silver Horde encounter samurai - all of these are obviously Japanese.
Rincewind also encounters peasants cultivating rice and wearing large, bowl-like hats; this is suggestive of China. But he also encounters Noh theatre (and Tsimo/Sumo wrestlers), which are definitely Japanese. And, of course, the book's title - "Interesting Times" - refers to the myth of the Chinese curse (although to count all the references would take a long time).
Perhaps the point is that Cohen and his Silver Horde are a parody of the Mongolian Golden Horde. The Mongols definitely invaded and captured China, in their case destroying the Jin and Song dynasties and subduing western Xia and the Dali kingdom (which was based in today's Yunnan province of China). The result was the establishment of the Yuan dynasty, covering Mongolia and China. (In IT, the various feudal houses are Hong, Tang, Fang, Sung, and McSweeney ("very old established family!")).
Korea had also submitted to the Mongols, who then tried invading Japan twice, but both invasions were repelled by
kamikaze storms. This set the limit on the Mongols expanding eastwards, and they turned west, invading and conquering most of eastern Europe (up to the border between Austria and Germany) and much of the Mediterranean, down to the border between the Crusader States (then in decline) and Egypt.
The Mongols were famously tolerant as far as religion is concerned, just as the Silver Horde are. The Mongol invasion of China was led by the legendary Genghis Khan, and Cohen is eventually crowned Ghenghiz Cohen - although Genghis Khan died halfway through the invasion of China, and his work was completed by his equally famous generals and descendants, Kublai Khan and Ögedei Khan. Cohen is, obviously, luckier.
So yes, the Agatean Empire - insofar as cultural influence is concerned - is a bit of a mystery. Perhaps it takes the form it does because the Yuan dynasty eventually became, culturally, a blend of Mongol and Chinese cultures and religions.
And there you have it. I hope it wasn't too boring for you, but I'm sure you know that I read a whole heap of history!