Probably for the same reason horses were chosen in the real world: Of the animals that can be domesticated, horses arguably make the best mounts. There are something like 150 real-world species of large land animals, and of these, only about a dozen can be domesticated. The rest might be captured and trained, or possibly raised from infancy and bonded with, but they can't be produced in quantity or bred for specific purposes.

Again in the real world, there are basically six traits that a species has to have to make it possible to domesticate. This is mostly summarized from Germs, Guns, and Steel:

1. Diet - you have to be able to feed it. Large carnivores generally fail this test pretty hard. 100 tons of grain will feed 10 tons of herbivores or omnivores, but those will in turn only feed 1 ton of carnivore. Horses can eat grass.
2. Growth Rate - If an animal takes 15 years to reach maturity instead of one, it's going to be cost-prohibitive in most cases. That's why we don't have Ape-ranchers.
3. Captive breeding - can't do selective breeding if you can't select who it breeds with. Some species just won't breed in captivity. Some need elaborate mating rituals or they won't conceive (ex: cheetahs). Some have complicated territorial requirements (ex: vicunas).
4. Personality - The animal has to have a disposition you can live with. Hippos will try to kill you just to see if they can. Grizzlies are prone to violent rages when frustrated, and psycho-girlfriend levels of jealousy. Zebras are irascible SOB's, especially as they get older - they'll bite you and drag you around a bit if they don't like something they imagine you did.
5. Calmness - they can't be too prone to panic. Gazelles have never been successfully domesticated because they will bolt blindly at the first sign of strangeness and either leap over their enclosure wall or bash themselves senseless against it.
6. Social Structure - There has to be some kind of social structure that a handler can take advantage of. All large domesticated animals have five features in common: a) they live in groups, b) they maintain well developed dominance hierarchies within their groups, c) those hierarchies remain fixed once established, d) they can imprint on the creatures around them while young and accept a human handler as part of the group, and e) they have overlapping home ranges rather than exclusive territories.
7. Suitable as Mount - On top of all that, if you want them domesticated specifically as mounts then they have to be capable of resting while standing, traveling long distances, carrying sufficient loads, moving fast, etc.

To make room for horses, you just have to select one of the domestication criteria that each one fails - or at least compares unfavorably to horses. Giant chickens might fail for the same reason as ostriches (bad basic skeletal structure for long distance load bearing), giant wolves might bond too closely with their handlers to ever raise for sale, etc.

The fantasy mounts will still be used if they exist - giant chicken races could be a thing people bet on, giant ants might be used for mining, and every military will want flying scouts regardless of the price. They will all be breathtakingly expensive of course; your typical farmer won't have one. But even nobles who can afford whatever they want will likely pick a horse over a giant beetle. Kind of like why you would choose to drive a car instead of a backhoe on your daily commute.

Probably for the same reason horses were chosen in the real world: Of the animals that can be domesticated, horses arguably make the best mounts. There are something like 150 real-world species of large land animals, and of these, only about a dozen can be domesticated. The rest might be captured and trained, or possibly raised from infancy and bonded with, but they can't be produced in quantity or bred for specific purposes.

Again in the real world, there are basically six traits that a species has to have to make it possible to domesticate. This is mostly summarized from Germs, Guns, and Steel:

1. Diet - you have to be able to feed it. Large carnivores generally fail this test pretty hard. 100 tons of grain will feed 10 tons of herbivores or omnivores, but those will in turn only feed 1 ton of carnivore. Horses can eat grass.
2. Growth Rate - If an animal takes 15 years to reach maturity instead of one, it's going to be cost-prohibitive in most cases. That's why we don't have Ape-ranchers.
3. Captive breeding - can't do selective breeding if you can't select who it breeds with. Some species just won't breed in captivity. Some need elaborate mating rituals or they won't conceive (ex: cheetahs). Some have complicated territorial requirements (ex: vicunas).
4. Personality - The animal has to have a disposition you can live with. Hippos will try to kill you just to see if they can. Grizzlies are prone to violent rages when frustrated, and psycho-girlfriend levels of jealousy. Zebras are irascible SOB's, especially as they get older - they'll bite you and drag you around a bit if they don't like something they imagine you did.
5. Calmness - they can't be too prone to panic. Gazelles have never been successfully domesticated because they will bolt blindly at the first sign of strangeness and either leap over their enclosure wall or bash themselves senseless against it.
6. Social Structure - There has to be some kind of social structure that a handler can take advantage of. All large domesticated animals have five features in common: a) they live in groups, b) they maintain well developed dominance hierarchies within their groups, c) those hierarchies remain fixed once established, d) they can imprint on the creatures around them while young and accept a human handler as part of the group, and e) they have overlapping home ranges rather than exclusive territories.
7. Suitable as Mount - On top of all that, if you want them domesticated specifically as mounts then they have to be capable of resting while standing, traveling long distances, carrying sufficient loads, moving fast, etc.

To make room for horses, you just have to select one of the domestication criteria that each of the fantasy mounts fail - or at least compares unfavorably to horses. Giant chickens might fail for the same reason as ostriches (bad basic skeletal structure for long distance load bearing), giant wolves might bond too closely with their handlers to ever raise for sale, etc.

The fantasy mounts will still be used if they exist - giant chicken races could be a thing people bet on, giant ants might be used for mining, and every military will want flying scouts regardless of the price. They will all be breathtakingly expensive of course; your typical farmer won't have one. But even nobles who can afford whatever they want will likely pick a horse over a giant beetle. Kind of like why you would choose to drive a car instead of a backhoe on your daily commute.