Although officially called "Mog" some official material calls him The Moogle or just Moogle, for example in the EU wallpaper. In the Japanese version of Final Fantasy XIII-2, Mog is referred to as "Moogle" and rarely called "Mog". His entry in the Datalog of Lightning Returns: Final Fantasy XIII suggests his real name is not Mog and that he doesn't know his real name.
Although officially called "Mog" some official material calls him The Moogle or just Moogle, for example in the EU wallpaper. In the Japanese version of Final Fantasy XIII-2, Mog is referred to as "Moogle" and rarely called "Mog". His entry in the Datalog of Lightning Returns: Final Fantasy XIII suggests his real name is not Mog and that he doesn't know his real name.