I own a Warmoth Swamp Ash build, maple neck P-J configuration, passive. I also own two Fender Japan PB 62 reissues. Basswood bodies, maple neck with rosewood boards. One has an active EMG, the other is passive with a Duncan vintage 60's pickup. I have a Squier P-j with a custom neck, and a maple body. That bass has active EMG's. And rounding it out, a 2014 Fender American Standard Precision bass. That just got a DiMarzio Relentless P-bass pickup.
ALL of these basses sound like a Fender Precision bass. Are there differences in the body woods, as far as tone goes? There's been a lot of debate about all of this. But I still say, if I recorded all of these basses just with the P-bass pickup selected on the P-J's, and compared them to each other, I know that it would be very difficult to identify which body wood you were listening to. I compared my Japanese Basswood body P-bass with LaBella Jamerson heavy flats to my American Standard with medium Fender 9050's and the tone was so close, I don't think it would matter much at all, recording or live.