Yes and no.
Horses are not naturally only able to be handled on one side. However, horses do not generalise, and a large number are not used to being handled on the right due to tradition so deeply dictating "handle the horse on the left". It comes from cavalry times when riders carried a sword on the left hip, as swinging the left leg over the horse also meant swinging the sword over and this could be unsafe. So, they would swing the right leg over, thus mounting from the left side.
The question above about left-handed swordsmen is interesting. As a lefty, I can tell you that it's only fairly recently that being left-handed was accepted as an okay thing. In the middle ages, it was seen as a mark of the devil. Left-handed people had their left-handedness beaten out of them more often than not.
But I digress.
Is there truly a "wrong" side? No. Is it unsafe to approach a horse you don't know from the right? Sometimes.
MAKORA THOROUGHBRED SPORTHORSES