I believe it depends on the version you get. I think for the GBA version they toned down his ye olde English-isms. In the SNES version it's so thick that he is some times difficult to understand (at least he was the first time I played the game as a kid).
Someone familiar with the various French versions here. Just to get this out of the way:
This game was never released officially in French on SNES, Final Fantasy 6 was supposed to in 1994 but in the end ties between Nintendo and Square worsened and NoE only got Secret of Evermore in Multi-3, and had to get their JRPG fix from Enix and Natsume.
Due to the pathetic situation of SNES releases in Europe, everyone there would just import/pirate the English version, so everyone was familiar with that one.
In 2009, Chrono Trigger DS got a brand new French localization alongside its English retranslation by Slattery (a German and Spanish translation were supposed to be included too judging from internal graphics like "the future refused to change", but eventually they included those in the smartphone versions).
Much like people like to bash Slattery's stuff as "dry", "literal" and "awful" for not keeping the same corny changes as a certain previous translation, some French fans were declaring the French translation "incorrect" because it was done from Japanese and as such didn't have ANY of the English name changes. Some "incorrect" names include Slash Flea and Ozzie, now following the original Toriyama weird food-themed naming conventions Sauce Mayonnaise and Vinegar (slightly altered for easier pronunciation in French, I think it was Mayonay?), but also Kaeru who should have apparently been in English as Frog instead of Gren. Some even accused the French version of "spoiling the plot" or "lazily removing his alias name" and "on top of that not grasping the concept of proper L/R romanization".
It's like Final Fantasy X German controversy redux, where some people complained from the German translation being too accurate to the Japanese script to the point it doesn't match the very liberally-translated English audio (by the same translation branch credited for Squall and Zidane's drastic mood changes between the PAL/JP versions and the US/UK version).
I'm actually starting to wonder if the "new French translation" Bregalad is talking about was based on the Chrono Compendium one, because that also has a load of untranslated Japanese honorifics and, IIRC, the "Kaeru" name.
For what it's worth, the 2015 French fan-translation is apparently based on the JP ROM with VWF hacks made for it from scratch among other things. They're making one for Final Fantasy 6 too if I'm not mistaken. I'd say they were probably exposed to the Japanese text and had to have a basic idea of what each text means (considering CT's script is rife with unused lines in the JP version which aren't even there in the US one, and that throws off the count quite a bit if they were just comparing script dumps).
Not to say they couldn't have made some shortcuts using that Chrono Compendium version's script (btw, that ROM is based on the US version ROM), but the previous French fan-translation kept "Frog" in English so it's not like it's conclusive proof they're a fraud (if they used Magus instead of Maou or Demon King I wouldn't say necessarily the whole translation is based off Woosley's script either).