No two actual P-basses sound the same really, but here's what we know for sure about a P-bass vs. the P-only pickup on a P/J bass:
On a passive P bass, the signal path goes from the pickup, through the volume pot, through the tone pot, to the jack. Very simple.
If the PJ bass is active, it sounds like the preamp more or less, so all bets are off on comparisons.
If the PJ bass is passive, and wired Volume/Blend/Tone, the signal path from both pickups comes together in the blend pot before heading to the tone pot, and it's possible that the blend pot may make a mark on the outgoing signal regardless of the pot's position. Think of it as similar to a faucet like this one:
^ When you have this faucet all the way to the "cold" position, the valve inside might still be allowing a tiny bit of the hot water through. It's hard to know for sure. On this P/J bass, maybe the P pickup soloed does sound different than if the J pickup weren't there at all, because of the effect of combining the signal paths in the tone pot. Or, maybe not. Its hard to know for sure.
But, if the passive P/J is wired Volume/Volume/Tone, the signal comes from each pickup to its own volume pot, then through the single tone pot and out to the jack. More like this kind of faucet:
In the V/V/T wiring approach, the signal path for the P pickup is pickup--->volume--->tone--->jack, which is exactly the same as it would be if the J pickup were not there.
So, on that particular bass, it's pretty much a guarantee that the P pickup soloed sounds the same as if that same bass only had that P pickup.
Exactly as in how the water coming from the "cold" valve on the second faucet, if the "hot" valve were entirely shut, would taste exactly the same as if the "hot" valve weren't there at all. There's simple nothing in the chain that would make it any different.