The Long Term Rental Agreement allows you to drive a new vehicle and change it regularly. What about the gray card needed to be in good standing in case of control?
The steps and the amount of documents to provide to the Prefecture to obtain the registration of his vehicle is not what we prefer. By changing vehicles regularly via the Long Term Rental Agreement, it could have been a shame. Fortunately for you, it is not so.
The registration certificate must be issued in the name of the owner of the vehicle. In this case, you are the tenant. It is, therefore, up to your renter to make all the administrative steps to establish the registration card. The registration card will be in his name, and you will appear there, but under the mention “Tenant.”

The cost of the registration document is often included in the rental price

The renter will give you the registration card or a copy so that you can drive in good standing with your new vehicle. The cost of the registration is also the responsibility of the renter, and the latter has likely reflected in the rental price.