The Calvary carried their scabbards on the left so they mounted from the left so that they wouldn't have to grab their scabbard and swing it across the horse. It just became habit. Any horse can and should be taught to be mounted/dismounted from both sides. An aside to this... I was told that a Calvary troop saved themselves from an ambush when they noticed the "troop" they were meeting mounted from the right.
It was because soldiers carried their swords on their left, so they mounted on that side that way their sword would stay down against their leg. It is just a learned tradition, you should really practice mounting from both sides so that your horse doesn't become shy on one side :)