Well it's wired in series, but that's not what makes a P sound like a P. Nor is it the placement. You can put a J pickup in the P location and it won't sound like a P. You can also put a P pickup in the Jazz bridge location, and they sound very different. You don't get that percussive burp from the P in the bridge position. It's rounder and smoother sounding. The P has two coils, but it's not two pickups. Two pickups such as you would find on a Jazz bass both pickup the same strings, so they alter the tone of one another. The two P units never pickup the same strings. if you make a big single coil shaped like a P (such as some old Guild basses had) it would still sound like a P pickup... only not hum canceling.
The Precision pickup has a very shallow coil, and is wider than the Jazz pickup. Also the length of the coil is obviously shorter. This is what makes the P sound like a P. There's also a different amount of wire wound on each pickup, and having two smaller coils, as opposed to one larger coil changes the tone as well.
Another example is the difference in tone between a Strat pickup and a Jazzmaster. You can see the obvious difference in the shape and size of the pickups. The Strat has tall skinny coils, and the Jazzmaster has short squat coils. The Jazzmaster pickups are very similar to P bass pickups, and get that fatter hollow tone.
So the bottom line is it's the shape of the pickup that makes the sound.
Wiring two J pickups in series only gives them more bottom and less top... but that's not really what a P bass sounds like... that's just one sound you can get from a P bass with the tone turned down. Listen to John Wetton on any of the King Crimson albums he played on (such as the 1973 "Larks' Tongues In Aspic," "Red" from '73, and "Starless And Bible Black" from '74)... that's what a P bass sounds like!
In the 60's and 70's (and even 80's) not many people played Jazz Basses... most of the recordings you hear are P basses, and they don't all have that thick dub tone. The Jazz bass craze just started recently. Personally I'm sick of hearing Jazz Basses and SVT's... no one has an original sound any more!