the white horse is probably a warhorse, and I think that one of the races can get spider mounts, but have not played as it only seen it in the diplomacy window.

harpo

The white horses can be also obtained from the shop of a city that has grown to absorb a stable. 

As long as the owner side has researched the technology, it does not matter whether it has the "Legacy of SomeoneOrElse" trait.

So if you want to equip your heroes with a warhorse (significantly better than a horse) you have to.

1. Find a tile with wild horses.

2. Build a city nearby

3. Snake the city to reach the tile

4 Research the mounted warfare tech

5. Improve the tile

6. Pay for a horse in the store

Note that you cannot give warhorses to your trained troops unless your side has the "Legacy" trait.

Try the wargs, too.  They are usually better for mages and archers. 

A city that has a stable INSIDE the city limits?  Not in the city's zone of influence, inside the city's walls.

I am not aware for an analog for wargs.  There are plenty of other mounts - boars, spiders, etc... but I do not think you can get them in any way except finding them in quests and in the possession of heroes for hire.