I have an 09 Precision, a 76 Precision body modded into a PJ with an 83 neck, and a Hammersmith PJ4. All three of these basses sound way different unplugged. But they have the same strings and pickups, and when you turn on the P pickup alone in the PJ's, they all sound identical on a recording. I also have two Guitar Factory basses that are very dissimilar except for the pickups and pickup locations, and one is even a 5-string. Unplugged they couldn't sound more different. Plugged in, they sound identical. But my 95 P Lyte with the same pickups does sound different both unplugged and plugged, however, it's a radical departure build-wise from the GF basses with a basswood body and bolt-on neck, whereas the GF's are neck-thru and have the same mahogany body woods, although different size bodies.
So what does that tell me? It tells me that the build has to be radically different for it to matter, and with similar pickups, similarly built basses will sound sound similar, even if the unplugged sound is different.