Yeah it was bench seating so one guy had the reins and the other had a shotgun. Hence the name.

The apocryphal story is actually kind of interesting.

Roads and right of way established during the pre-firearm era were that you'd ride on the left, with people going the opposite way on your right. This was so you could use your dominant hand (usually your right) to use a sword to defend yourself.

Roads after firearms were available often established right of way with riding on the right, with oncoming traffic on the left. This is because when you shoulder a firearm on your right shoulder it's easier to aim left.

Stagecoach drivers would sit in the left seat, with the extra person sitting on the right, holding a shotgun, hence the colloquial term for the front passenger seat.

I have no idea how true this is, but it makes for an interesting story.

I'd been told it was a gangster thing: passenger seat shoots out the window for a drive-by.