I wouldn't say that is entirely true. I have a V63, so maybe it's way different than the 4003, but the Bridge pickup solo'd, wide open is what I would call in the same ballpark as as Precision run wide open with roundwounds. Now a Precision won't sound like a Ric in any other pickup position though. The Ric is the more versatile bass in the sense what you can get a larger number of varying sounds out of it. Maybe it'd be best to compare a Precision to a 4000, just single pup to single pup. Identical, no, they each have their own distinct voice, but not worlds apart.
I am ordinarily on the bridge pickup and wide open on my 4001. It sounds absolutely nothing like a P bass to me.