I had a 375 for a while, and apart from the wide fretboard which I couldn't get on with, it was a great bass.
Physically, the 4 will play more like a P with the wider fretboard at the nut. Tonally, though (and that's what really defines 'P-ness' or 'J-ness') it's like neither, as the pickups are completely different in nature to either P or J pups.
The pups on the Yammy are side-by-side dual-coil humbuckers.
The pups on the Jazz are single coil non-humbucking (like a Strat) and when both are on are in parallel.
The pups on a P are two small units designed to be humbucking in combination and wired together in series.
The Yammy neck pickup gets somewhere near the P sound but not exact, whereas the humbucker at the bridge, when solo'd has a sound all of its own.
Personally, I like flexibility and it would have been nice to be able to have switched the pups to single coil, but I really liked the tone of those pickups.
G.