Can a dual Schengen/non-EU citizen enter the Schengen area with his non-EU passport?

There is no rule like that in the Schengen regulations so, as far as EU law is concerned, it is not mandatory to identify yourself as an EU citizen when crossing an external Schengen border. Theoretically, there could be some rules like that at the member state level (e.g. it could be mandatory to present a French passport when entering France) but that's not very common in Europe and definitely not the case for France.

However, if you present a Japanese passport, you should have a valid purpose for the trip, sufficient financial means for it, etc. and could be refused entry if you don't (which would not happen for French/EU citizens). You will also get an entry stamp.

Note that France is fine with multiple citizenships but the situations in which one could be both a citizen of Japan and of another nation are really limited. If you know a French citizen who also holds a Japanese passport, chances are that they actually lost their Japanese citizenship a long time ago or risk losing it at any time, even if they have been able to hold onto their passport and/or renew it in the meantime.

This makes the entry and exit stamps all the more important: If you are French, you don't risk any serious trouble with the French or Italian authorities but an inquisitive Japanese border guard or passport officer might notice a dangling entry stamp or a trip to the Schengen area without matching entry/exit stamps and realise you actually have another citizenship at which point you could be formally asked to choose between trying to renounce that citizenship and giving up the Japanese citizenship.