Some countries give nationality by birthplace (e.g. USA), some by parentage (e.g. Sweden), some by both. It really depends.
BTW having the actual passport does not change status. That is, you could be a citizen without the passport. It is just a proof of citizenship for travel purposes.
According to French laws baby can automatically become a French citizen as long as one parent is French. Only thing you need to do is have the father go to the French Embassy with his own French ID card (or even better his birth certificate as passport is not enought to prove that he is a French citizen) and the baby birth certificate.
I would suggest that he also takes some passport photos of the baby with him so that he can apply at the same time for baby's passport and ID card. Only the French parent can apply for those documents as he/she will need to sign the documents on behalf of the baby.