No, the remote fob does NOT record CEL (OBD II aka check engine data)…nor other “warning lights. It does keep mileage (odometer) & VIN data…as well as CBS (condition based service) info, though.
But the car does keep fault code data. The CEL (OBD II) system is a federal requirement for recording EMISSIONS SYSTEM data which has been mandated since 1996. The car also keeps manufacturer fault codes that are above & beyond the OBD II emissions mandate. This info can be retrieved through the diagnostic port connector.
You can see info below in regards to the DDE (digital diesel electronics) system and its communication bus.