I think a main reason for that is that Fender has made so many different P pickups, and that's why I get a little confused when people talk about the "P sound". Factor in the tonal variations from different strings and the P sound gets even broader.
I'd really like to see list of the details of Fender P pickups over the years (sc's included) that gives the year, DCR/turn count, coil height, wire gauge/type, magnet type/height, etc. That would be interesting; you could look at the specs and get an indication of how a particular pickup may sound, and you could also use that info to guide you in pickup selection if you've decided on a tone you'd like to get from your bass.
I also agree with what Jim said about necks from my limited experience with swapping stuff around.
+1