I always see/know of European cheeses, but I've never heard of good Chinese or Eastern cheese...but they had cattle and milk. So why is it I've heard of fancy French cheese but not fancy Thai cheese?
In Korea, the particular breed/s of cattle they had produced very little milk. Cows were kept for work, nothing else.
According to the annals of Chosun (조선왕조실록), when a king wanted to drink milk every day the court rose up in defiance, calling it an inhumane act to take mother's milk from calves. The king relented.
It's not to say that Korea doesn't have food that use milk, as goats did exist and produced some milk. But most recipes were something that came from Mongolian influence, and not staple.
Cheese and dairy products were not traditionally part of the diet of many of the lowland agriculturlaists of East Asia, including most Han Chinese diets. Milk, yougurt, ahd cheese were part of the diet of East Asian nomadic pastoralist, like the Mongols, Kazakhs, etc, but we don't have much cultural interaction with them in the modern West.
However, cheese and dairy products are not an exclusively "Western" thing if we include the Middle East, North Africa, and Iran as "non-Western". There's a whole variety of local cheese that are manufactured in Turkey, Iraq, Syria, and Iran, many (but not all of them) of them using sheep or goat milk rather than cows milk. The greek feta that you might be familiar with is representative of some of these styles of cheeses.
My favorite kind of Syrian-style cheese is jibin, which can be found in ethnic markets in Europe and America, but is really best if made at home with whole milk. It's a soft firm cheese, made by heating whole milk in a saucepan with rennet and salt. Once it congeals, you let it cool, then press it in to hamburger-sized patties with your fingers. If intended to be stored for a long time, you will sometimes put in a jar of oil with spices, or in a spiced brine.
This is an indirect and incomplete answer as I started reading and never finished this book:
The author talks about different milk cultures in different parts of the world, some were more cheese oriented, some more milk oriented, some more yogurt oriented, and the different kinds are generally geographically connected. What kind of dairy culture (ha! culture~) developed was partially a result of climate. Hot humid India became a yogurt culture, for example.
You'll find plenty of dishes like milk noodles, fresh yogurt and yak cheese in China as you move closer the historically pastoral West and Tibetan areas (saw this in Xining). Unfortunately what we see of Chinese cooking in the West is primarily derived from Fujian, Guangzhou and Taiwan. These are all excellent cuisine traditions but fail to bring anywhere near the diversity that is spread throughout China.