In medieval time, different religions coexisted in China, Japan and Korea, but only Christianity was allowed in Europe. What factors made medieval East Asia more religiously tolerant than medieval Europe?
When Buddhism was transmitted to Japan, for example, the local population have no trouble reconciling Buddhist doctrines with their native Shintoism. The Shinto kami were integrated into Buddhist cycle of death and reincarnation, while the Buddhist deities were incorporated among the kami. Thus, although Japan had a strong national religion, its inclusive nature allowed it to accommodate the importation of Buddhism.
In China, the belief system was dominated by Confucianism, ever since it was adopted as the national philosophy by Emperor Wu of Han. Confucianism prescribed many near-theological tenets that demand religious observance, but makes no real statements on the existence or non existence of other deities. It was happy to co-exist with both Taoism and the later introduction of Buddhism - as long as there was no conflict of interest.
Note that in both China and Japan (since you tagged those countries), persecution of religious groups did occur. For example, the Three Purges of Buddhism in China. Purges took place in response to perceived political threats or for other political reasons. Emperor Wuzhong of Tang imitated a purge to increase tax revenues (they were tax-exempt). In Japan, Toyotomi Hideyoshi ordered the expulsion of Christians because he was feared the Roman Catholic organisationa and the possibility of religious revolts. He might also have been concerned about reports of Japanese individuals being abducted and sold abroad as slaves.
An incomplete but in my opinion a large part of the answer summarized by myself from the book written by American lawyer, writer, and legal scholar, and John M. Duff Jr. Professor of Law at Yale's Law School Amy Chua book Day of Empire: How Hyperpowers Rise to Global Dominance—and Why They Fall:
According to Chua, Genghis Khan was a "Defender of Religions" and The "Golden Horde" of Genghis Khan and his descendants was famously religiously tolerant as they believed in a 'Cover all your bases' strategy regarding God/Gods and they slaughtered anyone that did not consider the Mongols as the supreme authority. Genghis Khan was a "Defender of Religions"
The Mongols took all of China. They destroyed the Jin dynasty, the Xia, the Dali Kingdom and the Song. They killed over 45 million people (I've seen much higher estimates) in China.
They moved south and subjugated the people of Vietnam/Korea/Cambodia
They they moved West and the Mongols destroyed the Khwarezmid Empire and then devastated and depopulated Baghdad and the surrounding areas so thoroughly that many historians don't consider them recovered even today.
Then they went north and killed about half the population of Kievan Rus according to Parallel Sixty - a website concerning Russian history.
To summarize, the Mongols ruled >25% of the populace on earth (and killed 1/4th to 1/10th as many) in the exact area which was inquired about for a period of ~200 years which was chronologically located in the middle of the also referenced Medieval era and didn't participate in any holy wars or tolerate them in the slightest. Should you be a vassal or were belligerent towards allies or acted unacceptably then they destroyed your society and institutions (including troublesome religious institutions) completely and then punished the populace harshly.
East Asia is more "religiously tolerant" than Europe in medieval time because of several reason.
- The ancient east Asia is agricultural economics system. Agricultural Economics heavily rely on environment such as climate, land, water supply...... Therefore the worship of agricultural gods and nature gods has important position in early east-asia religion. In additional, those gods have different "role" and Polytheism are not contradicted.
- The east Asia society is patriarchy. In patriarchal society, the ruling class is the oldest man within the group and all experiences, techniques, values and wealth are transferred within family. Therefore ancestor worship becomes another religion.
- The political system in east Asia is absolute monarchy. Obedience of patriarchy and nature believes will strengthen their ruling authority. Therefore , in addition to the original religions, other authorities including deceased ancestors, worthies (Such as Guan Yu, Yue Fei) and the emperor/king were also sacred. Japanese political system is different from other east asia countries but what the governors do are the same. "Religiously tolerant" are often used for the tools of domination. ZongBing (宗炳) in the book of immortal God(明佛論) said: 'Although Theory of Confucianism, Buddhism and Taoism are different, but their goals is same to rectify people.(孔、老、如來雖三訓殊路，而習善共轍也) Emperor Wen of Liu Song said "If everyone were Buddhists, I would no longer afraid of losing my emperor position."(若使率土之濱皆純此化，則吾坐致太平，夫复何事). You can see how political utilitarianism of religiously tolerant they are.
Based on the economical, social and the political reason, East Asia countries are said to be "more religiously tolerant" than Europe in medieval time - What I mean is, only the religions follow those rule can exists. Manichaeism, Nestorianism and Zroatrism actually are religions of foreigners and not much local Hans believe them. That is the same for Christianism, even though it was developed quickly in late Ming and early Qing dynasty, it was banned in early 18th century due to contradiction of royalty and local practise. In Japan conflict/war between the local religions and Christian were common in Azuchi–Momoyama period.
However in 19th century it changed. The nature and agricultural worships are being replaced by Science, the patriarchal society was collpasing due to urbanisation and industrialisation and the royalty was threatened by riots and western invasion. Box rebellion is a classic example of the religious conflict in China, though it occurred in 1900.
Asian philosophy is inherently more about personal freedom, Buddhism, The Veda's Taoism, all stress personal freedom. The lack of religious tolerance was and is much more political or economic in nature. When some find the means to control others with a particular religious belief then intolerance sets in as a way to maintain that control. Even Christianity was originally more about personal responsibility then about a place to worship.