Mounting a horse from the left is purely because you carried your sword on your left side in its scabbed and drew it with your right hand. It’s impossible to throw your leg over a horse with a from the right with a sword on your left hip. That’s also why they rode on the left of the track. You held your sword in your right hand and it was the natural way to fight and more difficult to reach over your horse and hold the reins.
The horse doesn’t care what side you get on.