On older Fenders, it's below the strings, later ones it's above, then it disappeared entirely.

It's a finger/thumb rest.

Back in the 50s/60s you'd rest your fingers there & actually play with your thumb. I've never tried it, it's just too weird for me, though I used to own a Jazz bass old enough for it to be placed there.

Later it became more fashionable to play with your fingers, so it was moved to above the strings as a thumb rest.

As styles changed - more people used picks, others realised you could just as easily rest on the pickups, it just went out of fashion & many basses no longer have one at all.

Similar happened with the pick-up guards - or 'ash trays', as I used to call them. They were there to protect from stray interference, hum etc, but as people moved away from that early fixed-position playing style, they increasing were just in the way. To start with, people would take them off themselves - considerably easier on a Fender than a Rickenbacker; a couple of screws, vs take the strings off & the entire pickup out. I eventually lost my 70's Rikki ash-tray after a house move, but it was one of the first things I did when I bought it, new.

On older Fenders, it's below the strings, later ones it's above, then it disappeared entirely.

It's a finger/thumb rest.

Back in the 50s/60s you'd rest your fingers there & actually play with your thumb. I've never tried it, it's just too weird for me, though I used to own a Jazz bass old enough for it to be placed there.

Later it became more fashionable to play with your fingers, so it was moved to above the strings as a thumb rest.

As styles changed - more people used picks, others realised you could just as easily rest on the pickups or the edge of the scratch-plate, it just went out of fashion & many basses no longer have one at all.

Similar happened with the pick-up guards - or 'ash trays', as I used to call them. They were there to protect from stray interference, hum etc, but as people moved away from that early fixed-position playing style, they increasingly were just in the way. To start with, people would take them off themselves - considerably easier on a Fender than a Rickenbacker; a couple of screws, vs take the strings off & the entire pickup out. I eventually lost my 70's Rikki ash-tray after a house move, but it was one of the first things I did when I bought it, new.

I found someone did a blog page just filled with images of different rests, factory & custom - https://goleilamccoy693.blogspot.com/2018/03/fender-p-bass-thumb-rest.html