Thanks to Ian for pointing out where the French custom might come from. Here is what the museum of hoaxes thinks the origin might be:
"The linkage between April Foolery and the Springtime is seen in another story that traces the origin of the custom back to the abundance of fish to be found in French streams and rivers during early April when the young fish had just hatched. These young fish were easy to fool with a hook and lure. Therefore, the French called them 'Poisson d'Avril' or 'April Fish.' Soon it became customary (according to this origin theory) to fool people on April 1, as a way of celebrating the abundance of foolish fish. The French still use the term 'Poisson d'Avril' to describe the unfortunate victims of April Fool's Day pranks. They also observe the custom of giving each other chocolate fish on April 1."