In practice, no. The French registration certificate (certificat d'immatriculation, still called the carte grise) is issued through the ANTS online system and shows the holder's French address. Every application needs a proof of address in France less than 6 months old: an energy or phone bill, a tax notice, a rent receipt, a lease or title deed, or a home insurance certificate in your name.
On top of that, a used car brought from another EU country needs a certificate of conformity, a contrôle technique less than 6 months old if the car is over 4 years old, and a quitus fiscal from the tax office showing VAT is settled.
If you have no French home, the process stops at the address step. Using someone else's address or faking a lease is not a shortcut: false documents are a criminal offence and can give your insurer grounds to refuse a claim.