A fantasy staple, the fantastic mount (TvTropes). People in Fantasy Land ride giant yellow chickens, giant wolves, giant beetles, caterpilars, ants, and other insects, evolved bears, sharks, miniature dragons, wyverns, pterodactyls, and a plethora of other creatures.
But the horse, the ordinary, commonplace, Equus ferus caballus remains present. Why do people still use horses when there's so many other choices?
I know that video games make it appear that the only difference between a horse, a dragon, and a giant spider is cosmetic, but this is a fatally simplistic perspective. These creatures are very different from each other, and when you make them bigger you also make their drawbacks bigger.
Also, every one of these creatures will produce a lot of gross waste. If you think a pile of horse dung is nasty, just wait until you see the revolting slurry that sprays out of a pterodactyl.
Horses are safe. Horses are good. Horses do not silently kidnap neighborhood children, wrap them in silk, suspend them from the roof of the barn, and let hundreds of little horses literally suck their liquefied organs out for nourishment.
A reasonably well trained horse (not the "groom killer" type that some macho riders have sometimes preferred due to a delusion that they're stronger or faster than a well-disposed steed) is loyal enough to make a good dog look like a fly-by-night, will uncomplainingly let a human ride it to death, is about as smart as a dog (maybe smarter, depending on individual variation) -- and far easier to stable, train, and care for then a giant yellow chicken, dire wolf, etc.
Horses are likely cheaper to feed than most fantasy mounts, as well. Although a horse needs oats (or other grain) to give maximum riding performance, when not being ridden regularly horses can live pretty well on less expensive hay or (nearly) free open grazing. Horses have been bred on a routine basis for centuries, while these special mounts are presumably relatively new to the task (else they wouldn't be "special"), and the skills to breed, care for, and train them, as well as to make and use their accessories (saddle, harness, tack, etc.) are common, rather than exotic.
Fantasy Mount
Horse
Not everyone drives a dragster. To buy bread a beaten up Honda Civic is more than enough.
Same for your mounts. Why messing with a monster when a horse is quieter, safer and better manageable.
Magic is very intensive on the body. It requires a lot of fuel to make whatever unnatural body plan work. As such, you need magical food. This food is generally quite expensive and intensive to grow.
Horses eat normal food. As such, they are the reliable choice if you can't source whatever fancy exotic stuff your magical mount needs.
I'm going with a more emotional approach that can be the basis of logical reasons. Humans can have a special bond with horses and it can lead to a lifetime companionship. The relationship is marked by mutual trust, respect and affection in which humans and horses would see each other as family. Horses are emotional and social animals and they can suffer from loneliness just like us. Even in the wild, horses are pack animals that tend to stick together and have a family structure. This bonding would be the main reason why everything else would work out well.
Humans can't bond with other mountable fantastic beasts like horses and they are not as tameable as horses. With horses, people can also form a cavalry per the pack structure and bonding; where there is strength in numbers.
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Would you take a tank, a tractor, or a helicopter to go to McDonald's?
No?
The same applies here. Giant tigers are awesome for fighting. Rhinos are epic landmowers. Dragons fly you anywhere in no time. But if you're just going to your daily office work, you just need a mount that's fast, reliable, and small enough, without need for special care.
Horses are perfect for this. They won't eat John, the hostler. They won't rampage away randomly, demolishing the pens and then everything in their general direction. They won't burn down the office when they sneeze. They just munch away calmly.
Just as in our life you ron't take a dragster or a Cenessa to go to work, you won't need special magical animals for your daily commute. Every animal breed will have its place in this world, and horses will be your regular Honda Civics.
I blame the labor unions.
The union representing pegasi has won time schedules that keep pegasus operators from using them as regular mounts. 8 hours of rest after one hour of flight time. Only divine beings (and their representatives) can afford the stabling requirements necessary to meet work-rest hour requirements.
Dragons United demands a wage scale that frankly explains why dragons end up perched on massive hoards of gold. Hiring dragonslayers to eliminate dragons willing to work at below-guild scale helps to keep dragon mounts as something only the wealthiest (and soon to be former) billionaires can afford for now.
The demand by unicorns that their riders be (1) female and (2) either human or elven is currently under litigation in multiple jurisdictions for violating civil rights laws. Spokescreatures for unicorn societies have stated that they will quit the business totally if those requirements are overturned. Spokesorcs for Orcs Are People Two! have promised to continue to fight to overturn such restrictions as they are speciest as well as sexist.
Nightmares Unlimited™ is still committed to overturning health and safety regulations banning flaming body parts during the drought. They also claim that reports of luring young people onto their backs and hauling them off to Hell are fake news.
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:
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.
The benefits of the horse over fancy beasties fall into three main areas: it's easier to get, easier to maintain, and more broadly useful, even in a world where other options are common enough that they won't run into issues on that basis alone. While individual adventurers or heroes with unique use cases might ride a fantasy creature, everyone who is not a one-man army will find that the horse is the ideal, well, workhorse of a mount.
You can't just go to the mount store and buy a mount if you're in caveman times and stores haven't been invented yet. Capturing, taming, domesticating, and breeding animals is a sophisticated technology, and societies figure out how to do it for the horse long before they do it for anything else. So by the time society starts to domesticate something else as a mount, the horse is already in common usage.
After horses have been invented, they are easy to produce: their gestation period is relatively short and they grow up quickly. This is important because while a horse is still an investment, they are easy to find and to convince the current owner to part with.
You can stable a horse, and it will stay stabled, not fly away or decide to do something more interesting. As herbivores, they are easy to feed - critical for an army on the move. Horses are comfortable in broadly the same set of conditions that we are. They do not insist on being entertained, and will not mind if the rider tells bad jokes. In the cold, a horse is warm.
Horses are herd animals that are well-behaved in large groups (like an army). Horses are about the right size for a human to jump into the saddle quickly, and not to get hurt too much when they fall off. Horses are comfortable moving at a pace that humans find useful - neither crawling slowly nor shooting forward so quickly that they outpace the rest of the army. Horses can walk for as long as humans can. Finally, horses are heavy - in a cavalry charge, the horse's significant weight is transferred into the blow of the weapon in a way that a giant chicken simply couldn't match.
A horse is not just for riding. Horses will pull a plough or a cart. They are large enough to carry multiple riders. A horse's flank will shield you from sight or low-velocity projectiles. As a last resort, a horse is edible, and its remaining tissues are valuable for various crafts.
Depending on how magic works in your world, it may not be a good idea to be near strongly magical beings. You could be creating a world in which mana poisoning is a thing - throwing someone weak into an area of intense ambient magic would kill them due to their bodies being unable to handle it.
In a world like that, having a safe, mundane mount would be important for people who are not adventurers that have trained their bodies to be able to handle the intense magic coming off of strongly magical creatures.
Besides all of the reasons stated in the other answers, never understimate the power of personal preference and aesthetics:
"The supple muscles under the silky coat, the music of the hooves... the velvet of the muzzle, and the inscrutable black sea of their eyes. Ah, the horse!"
Lord Haurchefant Greystone of House Fortemps
Some animals are much easier to train than others, and are capable of forming a bond with their trainer or owner.
Not so very long ago, there was a guy with a Pet Hippo - that ended up killing him
5 years in, after being together for a long time, the hippo still killed him.
This is not an unusual sort of story - there's actually quite a short list of animals that are really capable of forming the kind of bond with a human that's necessary for them to be an effective and safe animal companion.
Additionally consider wolves vs. dogs. They're superficially pretty similar, but a significant shortage of pet wolves. Why? Their temperament, social dynamics, and acceptance of humans just isn't there in the same way. A few 'tame' wolves exist, but most of them remain pretty wild and prone to letting instinct take over.
https://en.wikipedia.org/wiki/Wolves_as_pets_and_working_animals
They're simply so much harder to work with, especially if you're looking to achieve functional domestication (e.g. able and willing to work with most humans, rather than just the one you know).
There's a reason there's never been any hippo cavalry regiments, despite how they'd be pretty amazing 'warhorses'.
Everyone is focusing on in-world reasons for horses being so common in fantasy settings, but can we take a minute to think outside the world? Having horses in a fantasy setting gives the audience a connection to the world. I know what horses are, you know what horses are, we've probably seen real horses in our real life. Not sure if I have seen any flying dragons or a person riding a shark in real life.
Commonality between your setting and real life gives the reader a grounding point to truly become immersed. Horses make a pretty good way of having reliable transport, especially when most fantasy settings, when looking at real world history, chronologically appear medieval. A setting that resists any connection to reality will likely be difficult for the audience to immerse themselves in.
