would it be possible to travel on a direct flight from Japan to France, and from France to Japan without owning any passport ?
No, you can't
For France, TIMATIC (document database for airlines)
France - Destination Passport
Nationals of France are allowed to enter with an expired passport. The passport is accepted when expired for a maximum of 5 years.
You could enter on a French ID Card (easier to get than a passport), but I don't know if Japanese exit immigration would be cool with that
For Japan, you need a passport either way
What I'd recommmend is getting a Japanese passport and doing that whole trip on it. Since they're a French citizen, they can't be denied entry and there are no worries of overstay, even on a Japanese passport, especially as there is a French parent that can attest of the child's French citizenship by descent by holding their French passport.