"What French ID carte? There is no valid ID cards you can get from France unless you are either French or not from the EU (carte de séjour/de travail)."
A "Titre de Sejour". You can request this even as an EU citizen.
I'm actually just renewing mine after an address change, Including all the digital fingerprint shenanigans for the "biometric chip".
It's not compulsory to have, but useful if you don't want to carry your passport around for ID.
(Which is compulsory at all times and doubly so when trying to get into most customer sites.)
Driving licences are not legal valid ID in France as it is not compulsory to update the address on them. Our UK passports don't have an address, but they are valid ID. Who knows how their warped bureaucratic minds work....?