It was not uncommon for people to add an aftermarket antenna mast like this to improve reception due to problems with the stock windshield antenna, especially in fringe reception areas. This is in no way anything from Oldsmobile. The ball at the base of the antenna and spring were common aftermarket antenna designs (the one on my 62 F-85 is exactly the same). The rear quarter was chosen because it was easier to get to than pulling the front fender. I have no idea why they picked the LH fender for mounting.

 

my old 63 dynamic 88 wagon had it on the drivers side.

 

Not sure what that has to do with an aftermarket install on an A-body coupe...

 

it was on the drivers side.i really can't remember ever seeing another car with an original left rear antenna.

 

Those that traveled to cities with tall buildings mounting it on the drivers side kept it father away from the buildings . Also less likely for the mast to be swiped by jokers on the sidewalk walking by while parked in the city streets. Yes even back then some had nothing better to do.