Yeah. Jazz. I wouldn't play one in a classic rock band, either. Sure, there are many songs that can be done with a P Bass, but many others require a tone off the bridge pickup. That's a jazz bass. Referring to your OP, the jazz bass is far more versatile than the P bass and, as such, is a bass far more suitable for general use. Sorry, but one-pickup basses don't cover all the bases. Of course, the American Deluxe Precision V would work just fine for most genres.
If you're talking about just playing the notes, sure, you can play the notes in any genre on a P bass. If you want it to work for the song, a P bass isn't going to work, tone-wise, in many songs in many genres. Try getting
this tone on a P bass (go to 8:00).
Or
this one.