If the cat is friendly, obviously used to people and to living in a house, and if it looks well fed and in good condition then it may well live locally and just be out on its rounds.
If it shows no signs of leaving then enquiries should be made in the neighbourhood – a notice (Avez-vous perdu ce chat?) with a photograph of the cat and your phone number pinned on notice boards, handed to local vets or dropped in mail boxes may generate a response.
If no-one comes forward then take the cat to a vet to be scanned for an identity chip. Any vet will do this free of charge and if he finds a chip he will contact the owner. A cat with a tattoo in its ear is most probably a feral cat which has been sterilised and released where it was found but the vet can check this too.
The Mairie of the commune where an un-chipped domesticated cat is found should be informed. If you cannot keep the cat until it is reclaimed or a new home is located, they will tell you where the nearest refuge or charity is that may take it. Many communes have a contract with a local SPA to take stray dogs and cats.
If you are able to keep the cat either permanently or until you can find it a home ask the Mairie if they have arrangements in place with any organisations to sterilise the cat as that will probably not have been done.
If the cat is quite scared and timid then it is probably a stray and almost certainly not sterilised. If you are feeding it at all then you must take responsibility for getting this done. The Mairie will know if there are arrangements with organisations to sterilise the cat. If no, look on Google under ‘associations chats’ in your department as many local groups will help with the cost and assist with trapping if it is too wild to be picked up. Once it is sterilised it would be kindest to release it near where it was found (cats are quite territorial) and to continue feeding it basic rations. It will certainly help control any mice or rats!
If you find a community of stray cats living together (the service areas on autoroutes are a common location) then take a note of the exact geographical coordinates to pass onto the Mairie of the commune and you can also contact one of the national organisations that have programmes to sterilise feral cats on a TNR (Trap, Neuter, Release) basis. One of the most active is https://www.30millionsdamis.fr/contact-2/