I'll stop short of saying "tone is in the fingers" because I think that greatly over simplifies the impact that the instrument, strings, amp etc have but what I hear is Janek Gwizdala getting as close as he can to his normal tone while playing a Pbass.
I haven't played the Fodera P, but I have played a bass with the Aguilar P pickup. It absolutely delivers the expected Precision tone. If anything it is very slightly modern with a bit more upper midrange (IMO) but it's far, far closer to a vintage Pbass sound than say, a normal EMG, parallel wound active P pickup.
But look at where Janek's hand placement is. How many guys pluck a P bass that close to the bridge? Almost everybody I see playing a P is either right on top of the pickup or shifted closer to the fretboard. Combine that hand placement (and technique - he's obviously spent years trying to be as articulate as possible) with new strings and the tone knob all the way open and I'm guessing it's not that hard to get most P's to sound similarly.
That makes all the sense in the world!
I don't think it is an accident that fusion/jazz virtuosos favor Fodera while a bass like Sadowsky is favored by sidemen and session players. Foderas seem to designed to capture individual nuance and technique above all else. I think a player like me whose technique is nuance free simply is not wired to get the most out of a Fodera. As always, a player has to know himself in order to pick the best instrument.