Just a very brief note from a non-specialist:
- Some kind of dairy products, such as 醍醐 (tíhú), 酪 (lào). and 酥 or 蘇 (sū) in Chinese, were known in ancient India, China, Korea, and even in Japan.
- One of the Buddhist scriptures, Mahāparinibbānasuttanta ('Great Discourse on the Final Nirvāṇa') that came to China and translated around ca. 400, mention some of them as following: 'Get milk from a cow, lào from milk, raw sū from lào, mature sū from raw sū, and from mature sū comes tíhú that is the best......(譬如從牛出乳 從乳出酪 從酪出生蘇 從生蘇出熟蘇 從熟蘇出醍醐 醍醐最上)'.
- These dairy products are also found in some 5th and 6th century Chinese agricultural textbooks, such as Qimin Yaoshu (Essential Techniques for the Welfare of the People) (link to the original Chinese) written by Jia Sixie in Northern Wei dynasty, and even Qimin Yaoshu had some recipes for them (氊酥酪乾酪法). Modern scholars as well as cooking experts have sometimes tried to re-produce these ancient dairy products, and some variants of lào are identified with sour milk, unmatured cheese respectively, sū with butter according to one of the latest attempt (Cf. HIRATA 2010). Nomadic ruling group often came to China and established themselves dynasty in China from the 4th to 7th century (Southern and Northern Dynasties Period), and the authors of the essay I referred to also point out some resemblance of the traditional dairy recipes of the Mongol- or the Turk nomads, rather than Indian original one.
- Korean (or Chinese) settlers brought milk and dairy product also to Japan in the 7th century, and emperor appointed some of the head of their family as special officers (乳長生) for dairy products (Saito & Katsuta 1986). These officers were responsible for delivering milk specially for emperor and her/his family. In Heian period, aristocrats also drunk milk and ate some dairy products. The dairy products were usually understand as something special or rather a kind of drug (supplement?) in Ancient Japan, and almost monopolized by elites such as Imperial family or aristocracy, and the commoners did have little access to them. Together with the decline of aristocratic culture, however, dairy production also seemed to obsolete in Japan: The last entry of such ancient origin pastures is dated to 1334, and a Jesuit missionary who came to Japan in the middle of the 16th century notes that the Japanese was not used to dairy products.
- In Edo Period, the Japanese still also understood milk and dairy products primarily as special food/ supplement, and the Dutch in the trading post in Dejima, Nagawaki, kept their cow to get milk (and a very small number of Japanese sometimes recorded the Dutchs' milk production in Nagasaki) (Matsuo et al. 2010). The spread of the large-scale dairy production in Japan, however, occurred after Meiji Restoration and mainly in the northern part of Japan, as is well known.
I don't have definite answer why dairy product did not became popular in (East) Asia, but literatures often points out the following possible contributing factors at least for Japan, rather than lactose tolerance, seemingly popular hypothesis in internet:
- Climate: Dairy products like cheese become easily decayed in such a humid and warm climate in East as well as SE Asia (Saito & Katsuta 1988: 353).
- Primary purpose of holding livestocks: the Asians are to have kept livestocks primarily as farmworks rather than dairy products, and their traditional cows were not so suitable for that purpose (Saito & Katsuta 1988: 351).
- Possible influence of Buddhism on keeping and making use of livestock and their product, but this popular hypothesis should be probably refuted. As we saw above, the earliest mention of dairy products are actually found in a Buddhist text.
The origin of modern (popular) cuisine in Asian countries itself should also be taken into consideration, but I know really little about it. Sorry.
References (sorry almost only in Japanese):