Names are granted to unnamed monsters by a person with high magical abilities. By gaining a name, the monster becomes a named monster and becomes much stronger than before, even evolving. For example, when Rimuru named the ogres, they evolved into oni. Many monsters become amazingly grateful to the ones that have named them.
However, naming a monster uses quite a bit of magical energy. Many people have lost all their magical energy after naming monsters, and even Demon Lords generally avoid it.
Sometimes naming can result in a permanent decreased in one's magical capacity. However, naming demonic beasts carries less risk than naming monsters.
If the master dies, a monster’s named status will disappear as well. Also, if it introduced itself with it’s real name, then that name can be used to manipulate it. Therefore, not introducing oneself was common sense among demon-kin.
Everyone in Tempest is a named monster.