according to carey nordstrand (and i'm gonna paraphrase here), they are different constructions: one is "short and squat" (the p) and the other is not. if you look at them side by side, one is definitely taller than the other.
to get a fuller explanation, you'd have to talk to someone who really knows this stuff, as i'm not really up on it all.
in my dual coils, i have one design (p coil) in the front coil and one (j coil) in the rear coil. there is a decidedly different tone quality between the two.
I think you're paraphrasing too much, or maybe you're simply commenting out of context. The reasons why a P pickup sounds so different from a J are many. One is typically a single coil single sensing narrow aperture design (J), and the other is a split coil (dual coils) dual sensing offset aperture design wired in series (vs parallel).
my guess is that the fodera (most?) dual coils are based on the jazz bass construction.
I would not make that supposition at all.
You don't design a dual coil pickup by just taking a single coil pickup and doubling it. the electronic model doesn't work out that simply. You start with tonal goals and start doing math based on wire gauge/material, number of winds, magnet size/type and gauss ratings, number of strings you have to span at what spacing, etc.
That's kinda' what I'm getting at...
What are you really trying to ask?
Are you just wanting to know if it'll sound just like a J bass in single coil mode? If so, I feel pretty safe saying no, BUT, it will have some of the same characteristics, which may be "enough". It is/was for me, anyway... and I'm mainly a J bass kinda' guy, tho I also currently own a P5 (one of Carey's, ironically) as well as several J and J-style basses (one is also one of Carey's as well). I've also done the PJ thing, but not the PP thing like some Spector 4 stringers.
But even within a true J bass pickup arena, per my earlier post, even within a fairly specific pickup type, the actual sounds can vary a LOT based on, again, wire, magnets, winds, etc. Carey's done a LOT with this (as have others), and one of his MANY projects with pickups was to find a way to fairly faithfully mimic a 70's J sound and a 60's J sound in a 5 string design (which is much harder than it sounds... you can't just add the extra length/poles... everything has to change).
Anyway, hope this helps.