But it probably is as old as roads. Also ships have similar rules.
The way I heard it, in the Middle Ages men riding horses would ride on the left in case when they encountered someone, that person was an enemy and then they would have their right hand (their sword-wielding hand) closest to the other man while trying to defend themselves.
According to that version, riding on the left was standard everywhere until the French Revolution, but the revolutionaries changed that because they were breaking with tradition in as many areas as they could. That custom then spread to many other countries, but not to Britain and its colonies (except Canada where it would be inconvenient to have the opposite of the US).