I don't remember this in the PSX original, I mainly used the JP scroll bug to gain JP but it stayed in that job alone no spill overs.
i went over to gamefaqs to see if this was a thing, i found this in the answer board:
"---SPILLOVER JP---
Whenever a character gains JP, all other allies ALSO gain some JP for the same
job, equal to 25% of the JP earned by the original character. For example, if
a character on your team who's a Ninja performs an action and gains 40 JP, your
other units will also gain 10 JP for the Ninja job, regardless of what their
current job is.
This bonus applies ONLY to characters participating in the current battle. It
does not apply to the other characters in your roster whom you did not deploy.
You also can't earn spillover JP if you're currently KOed. Other status
ailments still allow you to earn spillover JP.
JP Boost has NO effect on spillover JP. (It doesn't increase the amount of
JP that spills over onto others, nor does it increase the JP a character earns
via spillover bonuses.)
Note that "spillover" JP accumulates even for a jobs a given character hasn't
unlocked yet. You'll see it once the job DOES becomes available. For
instance, if Ramza earns JP as a Dragoon, Agrias will get spillover JP for the
Dragoon job, even if she hasn't unlocked it.
JP continues to be earned and distributed this way even if the character
already has 9999 JP for the job and his/her own JP count has stopped.
(I got it from the FAQ/Walkthrough of this game by Qu_Marsh here in GameFAQs)."
link to the page:
i just noticed that this is on the psp version board, so i doubled checked, here's a reddit link for the original: