What is a horse? A horse is a horse, of course of course!

Unless we're talking about types of horses.

Horses exist in many variations and with varying needs and levels of care. Think cars- Some cars are expensive. Cars in some regions are more expensive to own and drive. Cars in other regions are cheaper and easier to maintain. Your question kinda presumes equality across all horses, which is a problematic place to start. It's like asking, "Because a main battle tank is expensive, how do low-tech warlords field large numbers of trucks with mounted machine guns?"

Basically, it is comparing two things that shouldn't really be compared. But, I'll try to explain the differences:

Space

The first and biggest difference in cost is grazing space.

A nomadic army of Mongolians is something of a nebulous entity. While moving and on campaign, individuals and groups of warriors would spread apart, find supplies, and remain fairly independent until they would focus together for an attack. A single horse requires a large amount of grazing space to eat and maintain energy. Lots of horses require lots of space. The steppes have HUGE amounts of space. The Mongolian empire consisted largely of places that had lots and lots of space for their horses. The empire typically didn't stretch that far into areas that didn't have the grazing space for large numbers of horses. There are exceptions to this, but really, they did not move far into places without grazing space, or places where their style of horse-mounted combat on large, open spaces was not advantageous.

Europe is a lot more compact than a steppe. It is hard to maintain huge numbers of horses in forested, mountainous with lots of geographical barriers. Imagine trying to move thousands of horses through wood-pasture, or trying to set up a hit-and-run attack in a place where you could easily get trapped or flanked.

So, when you have your European armies with horses, you have to bring food for them because you might not find adequate pasture space, or you might be constrained in such a way that prevents you from leaving camp by a few thousand yards to find pasture for your horses while you camp for the night- if the army spreads a thousand yards across hills and forests and rivers, good luck reassembling that in any short period of time.

Horse type

Nomadic tribes, like Mongols, fielded large numbers of low maintenance and hardy ponies. Mongolian ponies are very roughly domesticated and have been bred for endurance and hardiness. They live very much like their (now exceedingly rare) wild counterparts. Really, a Mongolian horse and a wild horse have a lot in common, in terms of build. The neck, face, legs, body type, and size are very similar. They are not identical in confirmation, but they are close.

Western horses in Europe, the Mediterranean, and the MENA region have undergone significant changes over the last 5 thousand years. Many of these changes happened through focused human breeding programs. These breeding programs were designed to create faster, bigger, stronger, and more trainable horses. Breeding programs and breeding lines are generational projects, and things like stud-books and breeding guides were handed down through many generations of horse breeders.

Because of that kind of breeding program, horses started developing more specialized needs. Bigger horses require better food. Some start to need things like grain and oats. Because of changes in the horse's feet through breeding, some will need shoes. Their bodies start to become more sensitive to the elements, because their coats become more refined and less coarse.

And when these horses are trained, kept indoors, and become accustomed to human care, they lose many of their natural survival instincts. Taking a Prussian warhorse and leaving him on the steppes would be kind of like taking a Rottweiler and expecting him to thrive like a wolf (in some cases, dogs do adapt well enough to being feral, and horses did that as well- see the American Mustang. But, North American was something of an exemption due to exceptionally horse-friendly environment).

Mongolian horses survive very well off the land, and they are adapted for a hard, nomadic life. Western horses adapted for a life that did not require surviving in difficult conditions. This does not mean that Mongolian horses are "better." Western horses are capable of feats of athleticism and training that are far beyond the scope of anything one could expect of a Mongolian horse. An American thoroughbred moves faster than anything a Genghis Khan would have thought possible. But, to him, even Secretariat would be fairly useless because of the amount of specialized care a horse like that needs to maintain that kind of peak performance without breaking down.

Now, beyond that, MENA, Europe, and the Mediterranean are not always the most horse-friendly areas. Finding forage for that many specialized horses was extremely difficult, and many things relating to care had to be brought with the horse (if you spend a huge sum of money on a war horse, are you going to risk letting it starve?).

Horse training

Mongolian horses are, and were, very roughly trained. A Mongolian is interested in the horse as a very rough vehicle. When fighting, a Mongolian is most interested in riding without using his hands and shooting from a somewhat stable, raised position. The horse manages the speed, angle of travel, and deals with his own balance. The warrior might change directions (e.g. towards the enemy, away from the enemy), but otherwise, the horse was only roughly steered. This was a kind of horse riding well adapted to large, open spaces where precisely putting a horse in one spot did not really matter as much as covering distance quickly and effectively.

Europeans were MUCH more precise in their riding. This was because many Europeans cavalrymen used lances and swords. To get to the enemy, you had to ride into him. Additionally, you had to be able to maneuver your horse through a more compact environment.

Now, training a horse to run directly at the mass of point enemy sticks, banners, flags, loud noises, and general scariness took a lot of time and money. Training the European horse was a specialized activity, and it cost money. The war horse was an expensive tool because a huge amount of effort went into making one. Losing a good horse was extremely costly- you could not replace your trained warhorse with your average riding horse and expect similar results. And what results are those? A terrible, awful, powerful charge, with a thousand pounds of horse and rider crashing home at something at 20 miles an hour before scurrying away, turning back, and then hitting again and again, causing a massive hole in the enemy lines that you could exploit by rushing with infantry, or hitting with yet another line of heavy horse, causing the enemy to rout and run! Or something to that effect...

Method of warfare

Now, another part of the expense of western horses comes with the differences in how Westerners do warfare.

This is extremely generalized. A knight would have at least two horses with him at any given time, but likely more. He would have his "riding horse" and his "charging horse." The riding horse would be better suited for moving around on campaign. The charging horse would be better suited for battle. The battle horse would require somebody to handle him, and that person would probably be leading him from their own horse. Next, these horses require food, which requires some kind of draught animal, potentially a draught horse, pulling a wagon with supplies. All of these horses require food, grain, and other supplies to keep going. Additionally, the knight and all of his buddies required food, water, bedding, probably a change of clothing, additional lances, parts to repair armor, and other logistical items. Every knight increases the number of horses by at least three (but for a well off knight, at least 5 [In theory, a knight could have a procession of well over a dozen horses, belonging to him, his squires, his retainers, and his ancillaries]). This took a lot of space, and a lot of food, and as we established, the Europeans did not have the amount of pasture space available that Mongolians had.

A Mongolian warrior travelled with a string of horses as well, but his horses would be more generalized. They were more interchangeable between roles, and losing one, while definitely unfortunate, would not be quite as bad as a knight losing his extremely expensive warhorse. The Mongolians were responsible for their own horses, and they typically had fewer people assisting them with the logistical side of war.

Mongol armies also had little to no infantry, which typically require a more dedicated supply train of draught animals for food and supply. Because the average European soldier was not mounted on a horse, he could not carry quite as much in the way of food, and he could not explore as far as a Mongolian warrior to find food.

Specifically with the Mongols - first of all, their horses were more like ponies in size than the horses used in various periods of European warfare, which means that they were already able to field "more" horsemen than the former. Another reason is by way of their lifestyle, being a nomadic, pastoral people. There are a couple theories surrounding the anthropological reasoning behind why the Mongol tribes turned to pastoral nomadism rather than settling with agrarian farming (as was what many other successful civilisations had done, such as China) - one such reason is the geography of the Eurasian steppe, whereby a combination of lack of significant rainfall alongside the availability of wide, spanning prime grazing land almost paved the path towards pastoralism for the Mongols. Modern studies show that rotation of grazing land for certain livestock promotes growth of vegetation as well as preventing overgrazing, and this concept was not lost on the Mongols - to move their livestock they required mounts that could also be sustained off the land whilst being maneuverable, and so horses allow for a pastoral lifestyle to be maintained over many generations. The horse was so important to their lifestyle that it retained a symbolically important role to Mongolian culture throughout the years, and remains to this day - thus, unlike the Europeans, horses were a 'way of life' for Mongols (so much so that children were often "born in the saddle", learning to ride at an early age) and were essentially inseparable from their culture, and so off the bat we see how and why they were willing and able to field an expansive army based largely on horses. Horses also played a good role in warfare and hunting, of which the activity of archery within both formed another core "pillar" of traditional Mongolian culture. Also, their lifestyle allowed them to move through many thousands of hectares of steppe grassland, thus allowing them to easily (relatively) raise more horses to use. Tangentially, this article by National Geographic discusses historical meteorological evidence pointing to large, sustained periods of heavy rainfall prior and during Genghis Khan's rise to power over the Eurasian steppe, as it allowed for more horses to be bred and used by the Mongol Empire.

Good question. I am looking forward to the answer. As for Greece I would say the scarcity of arable land would mean only wealthy land owners could sustain a horse.