This is (as many other lovely forumites have said above me) a big part of it.
It is the sound of the bass on a whole lot of landmark recordings so it has, in a way, become the sound of the classic electric bass.
In my more modern experience, the P-bass - a decent one, not some piece of plywood sold at a mall for $100 - tends to sit perfectly in the mix and has a great blend of clarity and low end while not getting in the way of other instruments.
I’ve been asked to bring it - specifically - to gigs and recording sessions because that is the sound that the band leader/producer wants to hear.
I always bring another bass or two as well - just in case. Sometimes they are asked for - more aggressive, clean production jobs do well with a more modern active electronic instrument - slap and pop is better on a J-bass or MusicMan style bass - but for backing up a traditional band/song the P-bass has carved out its place as ~the~ electric bass sound.
Started with a P Bass in the early 80’ then moved on to a couple different J Basses as well as a few with Dual Humbuckers. Last year bought a Fender AM Pro II P-Bass, it has a 63’ neck that feels perfect to me and I absolutely love this bass and was the sound that I was looking for, apparently, I just prefer the tone of the P Bass more than the others.
Also measure the depth of the neck at the first fret, from the top of the fingerboard to the back of the neck.
The neck profile (curvature of the neck’s shape) is trickier to determine. It is likely some sort of a C contour, but might be a D. Hard for anyone not actually holding the bass to tell. Pics taken from a couple of angles would help.