I'm not sure about bike-related fines, but what I do know is that if you get an infraction (parking-related, speeding, etc) while driving a car on foreign plates (ie: Belgian, Italian, etc) most of the time they won't even bother sending it to your adress. Ofcourse this doesn't apply here since the OP gave the French rozzers his adress in Paris.
Oh, and the French Police were overreacting a bit with threatening to throw your friend in jail. While it is required to have at least one form of ID on you at all times within the EU, not having any ID on you will result in a fine, but nothing more than that (though repeated offenders may get harsher punishments).