When using a named nametag on a saddled pig the nametag is visually consumed and the player mounts the pig. The pig is then named with the nametag and the nametag is invisible in the inventory. When the player interacts with the empty slot where the nametag was it reappears (throwing it, placing an item on top of it on the inventory screen, or rightclicking with the slot, etc). When the player despawns the pig by travelling away it losses the name.