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What's the black box on a p-bass?

A rest. Either called a thumb rest or finger rest. I believe back in the day it was to park your fingers on while you played with your thumb. Lots of guys like to put one on the other side of the strings to use as a place to rest their thumbs.
 
That's the original position "tug bar", a place for you to rest your fingers while you play with your thumb, which is how they originally envisioned electric bass would be played...
Later, it moved to the E string side and became known as a thumb rest.
 
Don't believe these guys. They're fulluvpoo.

That "little box" is a tone block. You can get them made from various types/species of wood, and each different type will change the tone of your bass. You can even get one made from steel if you like to play metal.
 
It's a weird carryover from the prototypes, which actually contained a bubble level, so you could achieve perfect vibrational coupling to the stage.

But it all proved too costly, and honestly, it was an ergonomic nightmare, what with the vibrational coupling necessitating a neck-pointed-down-stance...(but now they explain it away as "neck dive"...boooo) but the conundrum is Fender had already produced THOUSANDS of pickguards of varying materials, as evidenced by your own photo, so the holes were already there. So what else could they do than screw in a piece of painted scrap wood and call it a "feature".... ;)
 
When you're sitting on your Fender stool playing, the bass always tries to slide out and down your leg that it is resting on at some point; it allows you to grab it and pull it back.

For all other basses, I have to keep my keys in my pocket so they can't slide out on me over the bump.

Nothing worse than dropping your bass while practicing on a stool.
 
They could fit in there because they didn't use batteries. They had to be wound like watches. The automatic winding models required that you move around on stage enough to keep it going.

That's actually what led Larry Graham to use slap technique-the percussive vibrations were enough to keep the mechanism wound through a set. Between sets, his tech would gently rock the bass to keep it topped up.