So, now there are Tengrist temples, and Tengrism is henotheistic to a degree. They also force Tengrism on their conquered Subjects. So Persia, Russia and a large population in China and North-West India are now Tengrists. How Plausible is this and what are the consequences?
Well, if the Mongols were to create and impose a more organized form of Tengriism, then the Mongol empire will be far, far less successful and far more destructive than it was in OTL.
The Mongols owed a good part of their success to their religious and cultural tolerance.
Many eastern Christians, Buddhists, and Muslims fought in the Mongol army, meaning that, without their OTL religious tolerance, the Mongol army will be a lot smaller.
And even without starting religious persecution, the Mongols were already pretty damn destructive in OTL, and there are quite a few cases in which the Mongols humiliated clergymen and destroyed religious buildings in order to break the morale of the local population.
To give a few examples of this; the Persian Muslim chronicler Juvayni, who actually goes out of his way in order to portray Genghis Khan and his followers as friendly and sympathetic to the Muslims, mentions how the Mongols humiliated sheikhs and imams in Ferghana by forcing them to tend to their horses, how the Mongols trampled copies of the Qur'an with their horses in [IIRC] Bukhara, and how they deliberately singled out the great Friday Mosque of Samarkand for destruction.
If the Mongols were to impose their religion on their conquered subjects, then we'd see stuff like this happening on an even bigger scale.