There's multiple reasons why the other societies couldn't utilize the horses in the same way as the Mongols did, some of which where the geography, the lifestyle, the training.
The area in which the Mongols lived and later expanded their territory was predominantly semi-arid steppe, thus an open terrain. That was perfect for the utilization of the horse as a war machine, for trade, quick traveling.
The lifestyle of the Mongols was deeply connected with the horse. The horse was the most valued animal, used for all sorts of things, and the children from the earliest age would be put on a horse, get close with it, and over time act like one.
All the Mongol children from very early age had an excellent military training mostly on horseback. The children were thought to be able to spend days on the horse, to be able to sleep on it, to eat and drink on it, to literally connect with it at the deepest level and as we would say to know how it breathes. The children were trained so well, that they were able to shoot arrows when the galloping horse had all of its four legs in the air so that they have maximum precision.
