I see at least some conditions:

• Docility (we can agree, that one of the worst things that can happen is that the animal will be disobedient or even aggressive)
• Speed (what is the point of slow "vehicle")
• Battle usability (obviously, if you need to use them in battles)
• Comfortability (this is probably not that important (just nice to have), saddles would be invented really quickly for any kind of animal and in battle, it is not the thing you are looking for).
• Special abilities (only I can think about is flying or stings)

From this, I think that a suitable answer would be...  cheetah.

• It is more docile than most of other carnivores. (see this link)
• It surely is fast.
• It is a beast, after all, so in battle, it can bite your enemies.
• It is similarly comfortable as the horses.
• It has no special abilities, but I think that animals with them are often less clever (birds and insects), so it would be very hard to train them.

I see at least some conditions:

• Docility (we can agree, that one of the worst things that can happen is that the animal will be disobedient or even aggressive)
• Speed (what is the point of slow "vehicle")
• Battle usability (obviously, if you need to use them in battles)
• Comfortability (this is probably not that important (just nice to have), saddles would be invented really quickly for any kind of animal and in battle, it is not the thing you are looking for).
• Special abilities (only I can think about is flying or stings)

From this, I think that a suitable answer would be... 

...cheetah

• It is more docile than most of other carnivores. (see this link)
• It surely is fast. And as was mentioned in the comments it would be great for shorter distances.
• It is a beast, after all, so in battle, it can bite your enemies.
• It is similarly comfortable as the horses.

...eagle, falcon (or something similar)

• Falconry is pretty common in some countries, so it is possible to train these birds.
• Usability in battle is also quite obvious.
• Flying!
• Comfortability may be an issue, I am not sure if it has the right constellation to carry a man.
• I am thinking about howl (because of night vision as special ability), but I still think that falcon is better.

...dog

• They surely are docile. We (as human kind) have a plenty of experience with training dogs for strange purposes. I am sure, that is would be also doable.
• Speed isn't always a strong point, but some dog breeds would excel even in that (sighthound)
• They are even some dog fighting breeds.
• Comfortability will be also no worse than when riding a horse.
• There are really a lot of different kinds of dogs, so if whole dog world would be bigger, there will be a good chance to choice the perfect match for every single rider (just imagine that).

I see at least some conditions:

• Docility (we can agree, that one of the worst things that can happen is that the animal will be disobedient or even aggressive)
• Speed (what is the point of slow "vehicle")
• Battle usability (obviously, if you need to use them in battles)
• Comfortability (this is probably not that important (just nice to have), saddles would be invented really quickly for any kind of animal and in battle, it is not the thing you are looking for).
• Special abilities (only I can think about is flying or stings)

From this, I think that a suitable answer would cheetah.

• It is more docile than most of other carnivores. (see this link)
• It surely is fast.
• It is a beast, after all, so in battle, it can bite your enemies.
• It is similarly comfortable as the horses.
• It has no special abilities, but I think that animals with them are often less clever (birds and insects), so it would be very hard to train them.

I see at least some conditions:

• Docility (we can agree, that one of the worst things that can happen is that the animal will be disobedient or even aggressive)
• Speed (what is the point of slow "vehicle")
• Battle usability (obviously, if you need to use them in battles)
• Comfortability (this is probably not that important (just nice to have), saddles would be invented really quickly for any kind of animal and in battle, it is not the thing you are looking for).
• Special abilities (only I can think about is flying or stings)

From this, I think that a suitable answer would be... cheetah.

• It is more docile than most of other carnivores. (see this link)
• It surely is fast.
• It is a beast, after all, so in battle, it can bite your enemies.
• It is similarly comfortable as the horses.
• It has no special abilities, but I think that animals with them are often less clever (birds and insects), so it would be very hard to train them.