Today, hanging out in Asia some more! Our question of the day comes from a little farther afield: Can you tell us some about the religion of the Mongolian Empire? As best I can tell the native Mongolian religion is Tengriism, but religious freedom and tolerance also seemed to be important to at least Genghis Khan. I'm not sure if the question is implying that Tengriism somehow isn't religiously tolerant, which isn't the case as far as I know, but I'm sure happy to talk about Mongol religion and myth!
Tengriism is a term that can be confusing in discussion of traditional Mongol beliefs; it's fairly modern, coined in the past few decades to describe the resurgence of traditional religion and its connection to national identities for Mongols and other Turkic people. But Tengriism as it is practiced now has very ancient roots in religion across inner Asia, and its deities, the eponyomous Tengri foremost among them, have been well-known and respected for many centuries.
We talk about Mongol religion and related mythologies as all being "Turkic", which refers to different peoples who share religious features and language families in common. There are deities and folkloric conventions in common among the Huns, Mongols, Hungarians, Xiongu, and Göktürks, who range from the Gobi Desert steppes to upper Turkey and all over the territories in between, although of course there are specific changes depending on what areas and peoples' specific cultural ideas are in play. Tengri (or Tangri, Tanri or Tura), the great sky father and creator of humanity, is the major god for most of these religions, but he's far from the only awesome deity involved; also appearing in the Turkic pantheons are the ferocious Mongol war-god Kisangan, the expansive earth goddess Etugen Eke, and Erlik, the god of the dead who is so awesome he has a dinosaur named after him.
It's part bird, part theropod, all ridiculous halfway-through-evolutionary-growing-pains, and was named after him because it was unearthed from the "underworld" that he administers in Mongolia. So that's excellent, but let's get back to the mythology.
Like many other religions in central and upper Asia, Mongol beliefs are strongly concerned with shamanism, most specifically with the shaman's role as the mediator between the spirit world and the mortal realm. Ancient Mongol shamans (and modern ones, come to that) were believed to be spiritually gifted so that they could communicate with spirits, gods and other unseen phenomena, which were the powers in control of good fortune, sickness, and the general cycles of the universe. Shamans were very commonly also considered the most important medical personnel because of the belief that illness was caused by evil spirits attacking humans; if they were angry enough no remedies could save the afflicted, and the shaman would have to journey to spirit world to beg them to desist or in some cases even enter into vicious combat to defeat them and drive them away. This was a super important and dangerous job, and it was understood that the shaman might very well be destroyed if they lost to a hostile spirit, which could tear apart and destroy their soul so that they never awakened from their trance.
The other most important religious role in Mongol tradition is that of the chieftain, the leader who represents their people and acts as their symbolic and functional figurehead and essence. This tradition is where our old friend Temüjin, a.k.a. Genghis Khan, comes in; he was not only the wildly successful war leader and political figurehead of his people, but also their literal representative for religious reasons. As far as the beliefs of his people went, his actions were the actions of his entire people and his successes and incredible power directly part of his people and from his people. As the symbol of all things Mongol (specifically the Borjigin clan, although his incredible success caused him to later become emblematic of all the confederated Mongol clans), Genghis Khan was powerful because his people were strong, and conversely his people were strong because he, their most important representative, was powerful.
This is one of the reasons that he is so popular now that he has his own cult, as a deified hero who became godly after his death. Mongol belief includes several classes of ancestral spirits who can be called upon, mostly by shamans but occasionally by common people as well, and Genghis Khan became one of them and was eventually promoted to almost the same status as the tngri themselves, something almost unheard-of since gods and ancestors are usually kept in separate hierarchies.
You're right that historically, Genghis Khan was said to be very interested in other religions. He almost couldn't help but be, after all; he was active during a period of time when Buddhism, especially from Tibet and northern China, was beginning to strongly influence many Turkic religions, and his expanding empire meant that he encountered many other religions as he conquered various areas and peoples, and needed to know what those meant to them and how they were practiced in order to integrate them into his empire. And that approach worked fairly well, all things considered; there was comparatively little in the way of active rebellion in Mongol-conquered territories thanks to this policy of pretty much letting people do what they wanted, at least when contrasted with contemporary conquerors that tried to institute religious overthrow.
I have to tell you about my favorite Mongol-specific myth, which is directly related to the idea of Genghis Khan's position as representative of his people and the general cultural ideas at play behind the idea of Mongol power and skill. A particular myth became popular during Genghis Khan's reign that claimed that in ancient times when humanity was first being created, the great blue wolves of the mountain caves came down from their homes and mated with human women, giving birth to a new breed of humanity with the fierceness of the wolf and the noble qualities of humankind. The Mongols believed themselves to be descended from these wolves, and saw in Genghis Khan the greatest expression of their lupine heritage, conquering the other races of the world as wolves always conquer prey animals. An alternative version has the wolves creating their Mongol progeny with sacred deer that represent the earth itself, making the Mongols the only human beings on earth who aren't, in fact, truly human beings, but something more.
That myth is the reason you so often see Genghis Khan referred to as "the wolf" or "the blue wolf" in various pop culture retellings of his life, and sometimes shown with wolves or wolf symbolism illustrating his link to this ancient ancestral idea.
This is like the most tiniest surface-pass image of traditional Mongol religion, so don't think it's even remotely a thorough vision of the ideas and cultural values at play; there are tons of them, as complex and rich and amazing and surprising as in any other religion, and scholars can spend their whole lives learning about them, especially since they remain living traditions on northern Asia to this day. As always, look it up if you're interested, because the awesomeness of myths and stories are their own reward!