I'll be danged if I can hear any difference. As far as I can tell a Fender factory pup will either sound good or bad, but era seems to play little in that, and maybe the bass strings are a thing that doesn't drive a pickup like a guitar string. There really is less nuance to pulling or picking a bass string. I say that being very aware that DiMarzios do sound a lot different, and other pups - like Duncan 1/4 lb stuff - can be designed to be unique. I don't like DiMarzios and that grindy overtone. I have personally used and loved Bill Lawrence vintage style Jazz pups, but I think you can more easily find a Fender 62 RI Jazz pup or fancy-pants Abigail Ybarra stuff that will all sound good and similar.
I do know that the early 80s 62 RI Jazz bass used the exact same pick up as the American Standard Jazz bass, just with a different cover and cloth leads.
