Well, according to the wiki:
- Mongolia: "Buddhism first entered Mongolia during the Yuan dynasty (thirteenth-fourteenth century) and was briefly established as a state religion. The cult of Gengis Khan, who had been accepted into the tngri, the highest pantheon of spirits in Mongolian shamanism, became annexed into Buddhist practice as well. Mongolia itself was at a political and developmental standstill until the sixteenth century, when after the conversion of Altan Khan Buddhism re-established itself"
- Buryatia: "The territory of the Buryats, who live around Lake Baikal, was invaded by the Russian Empire in the seventeenth century, and came to accept Buddhism in the eighteenth century at the same time they were recognizing themselves as Mongol; to which extent Buryat shamanism mixed with Buddhism is a matter of contention among scholars."
- Oirats: "The Oirats converted to Tibetan Buddhism around 1615, and it was not long before they became involved in the conflict between the Gelug and Karma Kagyu schools."
- Sarig Yogir: "The traditional religion of the Yugur is Tibetan Buddhism, which used to be practised alongside shamanism."
- Manchus: "Shamanism was the dominant religion of the Jurchen people of northeast Asia and of their descendants, the Manchu people. As early as the Jin dynasty (1111–1234), the Jurchens conducted shamanic ceremonies at shrines called tangse. There were two kinds of shamans: those who entered in a trance and let themselves be possessed by the spirits, and those who conducted regular sacrifices to heaven, to a clan's ancestors, or to the clan's protective spirits."
By looking at the map, Mongolia, Oirats and Buryatia are properly Tengriist, while Sarig Yogir is accurately buddhist.
Tengriist manchus are a bit of an issue, they were shamanist until turning to "chinese folk religion" a.k.a Confucian in the game (I assume) when they became rulers of China. In my opinion, they should start as Shamanists and switch to confucian when forming Qing.
Wiz, what was the historical justification into making them Tengri?