Cheese is very important in some Asian Cuisines, particularily among Turkic and Mongolian peoples.

The most known is kurut/qurut - hard cheese, with very salty taste. It is made of dried sour milk and is very popular, since it can be stored for a very long time, which is important for nomads. Then it can be eaten as it is or dissolved in water and drunken.

It's still very popular in Central Asia as a snack.

Let me add little bit more about importance of kurut in Kyrgyz cuisine. Unlike most nomadic peoples, Kyrgyzs live in mountains. For example, take Alay valley. Conditions there are very harsh and the only agricultural plant you can grow are potatoes, but they were virtually absent till the XXth century. Because of that traditional Kyrgyz cuisine consists mainly of meat and dairy products, in particular cheese.

Here are some pictures of kurut:

https://commons.wikimedia.org/wiki/File:Kasachischer_Kurt.jpg

https://commons.wikimedia.org/wiki/File:Fromage_kyrghize.jpg

The article in wikipedia: https://en.wikipedia.org/wiki/Kashk

Answering for India:

Cheese is created by adding the enzyme Rennet to milk. While alternative sources for Rennet are used nowadays, the traditional source of rennet has been from the Stomach / intestines of young calves. In fact, cheese is believed to have been accidentally discovered when stomachs / intestines of mammals were used for storing milk.

So considering that Cows were considered sacred in many parts of India, it is understandable that the technique for cheese creation would not have been stumbled upon. And even if they found out about it through contact with other cultures, the necessity for slaughtering calves to create it would make it not very popular.

I may be stretching your question a bit, but tofu (which is a very old food) is somewhat a form of cheese, as it is made by coagulating soy milk in the same fashion as dairy milk is coagulated to form cheese, however it usually uses calcium sulfate. There are also methods of preserving tofu. However, it is rather different in cooking behavior (notably, it doesn't melt) so the similarity between fresh mozzarella and tofu doesn't naturally spring to mind when you eat your big ole plate of mapo dofu, but it is the same basic idea of making a milky protein product into a solid food using coagulants. So the "idea" of cheese does exist in Chinese/Japanese cuisine.

In ancient Japan, while cattle were used as animals of labor, there was no tradition of milking cows. As the Samurai class developed, horses were given more importance than cows, and widespread use of dairy products simply did not emerge until the 18th century. Keep in mind the territory of Ancient and Medieval Japan, i.e. without the territory of Hokkaido, was not well suited for the kind of large scale cattle ranching practiced in a lot of places.

Source: 大谷元「チーズの起源と歴史（現代チーズ学）」、『New Food Industry』第49巻第10号、食品資材研究会、2007年10月、 25-36頁