I think the person you were originally asking was saying that the woods on a p bass and a j bass may be the same, but the change in tone comes at first from the different neck dimensions and body shapes. Like if you took a p bass and put a single coil pickup in it it would sound like a p bass acoustically and amplified. I don't have enough experience to tell that.I have a P pickup and at least one J pickup in the parts box so I could calculate the difference. I can't see it being anything more than a very small fraction of the overall instrument mass.
But I did just practice briefly with a Rickenbacker 4003sw (walnut body/maple neck/maple fb), Warwick SS1 (French Cherry body/wenge neck/wenge fb), and Warwick SS1 (American cherry body/maple neck/wenge fb). The Warwicks have the same strings and different fret/nut materials. The all wenge is the most growly with less brittle highs and more defined mids. The Rickenbacker and Maple SS1 are brighter but both sound different. Amplified, I went through a bunch of settings. They sound like themselves just colored a little different.