It is not the same thing at all. The US copyright is very different from the French droit d'auteur. The Berne convention merely fixes some common ground and minimum protection. For instance copyright can be fully transferred. The droit d'auteur is partly impossible to transfer (moral rights always remain to the author).

Copyright has a precise definition and is opposed to the droit d'auteur in its philosophy, so saying that France has copyright is a little bit like saying France has a king, sure it plays a somewhat similar role but is is fundamentally different