Basically, it's a combination of pickup location and simplicity:
Pickups closer to the neck sound fat and less punchy while pickups closer to the bridge sound thinner and punchier.
The P-Bass pickup sits right in the middle; not too fat, not too thin - punchy, yet still with some "bloom" to the notes.
Other pickups in that spot will sound pretty close to a P-Bass ('51 Precision, G&L L-1000, etc)
The other thing is phase cancellation - a bass with multiple pickups will have phase cancellation that scoops the midrange when those pickups are used together.
A split coil P-Bass has phase cancellation - but only between the two halves of the coil, which sense different strings, so the cancellation decreases noise without scooping the produced note.
That's the basic recipe - you can tweak it with string selection, body and fretboard woods, pickup selection and make it sound brighter, more thuddy, more open - but the basic sound of a P-Bass is a single split coil pickup in that spot.
If you want your other basses to sound closer to a P-Bass, turn their bridge pickups all the way down and just play with the front pickup.
It won't sound the same as a P-Bass, but it'll sound closer.