I send a Valentine’s day card to my girlfriend in Paris at 7 Rue de X, Paris, which is an apartment building: the residents each have a box in the lobby with the name on it. Turns out she lives in the building next door, 9 Rue de X. What might have happened to my card: would they have delivered it to a dead letter box in the other apartment building, or perhaps one at the local post office? I can’t imagine the card being sent all the way back to America. We want to find it, so what’s our best chance?
They shouldn’t have done so, but out of personnal experience, I know it happens. As you mention, normally, a mislabelled card would be sent back to you. But indeed, will they do so for a postcard from the USA??? I think your gf should pay a visit to her local post office and ask (or call the “centre de distribution” of her “arrondissement”, actually in charge of the mail) even though I’m not 100% conviced that what she will be told will be completely reliable.
Not a postcard, but a Valentine’s Day card. With pink hearts all over the envelope. And private notes 
Both countries are members of the Universal Postal Union, and are obliged to return mail. As a postal worker, I even see thousands of postcards with no return address that have been put back on a plane. These are clearly undeliverable, and I can only wonder at the carbon footprint of this practice, but there you have it.
The OP is basically going to have to rely on the apartment dwellers in the next building to get the thing back into the system (and having lived in apartments, I know most people are slack in this regard). Once it is back in the French mail system, it’s as good as on its way back to the sender. There might be a bit of a wait though - the only thing we treat worse than RTS mail is ex-overseas mail (so ex-ovs RTS tends to hang around a while in transit). Large countries like France and the US send each otherr enough mail that they keep tabs on one another (and arses are indeed kicked if things get too slow), but they might still push it a bit, because with foreign mail you can blame the other administration if it’s late, so domestic gets processig priority.
It could be sitting above the post boxes in the next door building. The “concièrge” or gardien usually leaves mail with a “new” name" there for a couple of days in case someone has moved in / is staying with a resident and they haven’t had time to update the name on the postbox. Then it will get sent back. It may be worth the girlfriend trying next door.