Often, there's a sidekick or a group of friends for the captor. They can play different roles like helping the captor in battles against those who want to take Darkrai, or providing moral support. And sometimes, there are mysterious characters who seem to have a connection to Darkrai's past and their appearance adds more mystery to the story.
In many of these fanfictions, there could be a rival trainer. This rival might be jealous of the captor for getting Darkrai and try to steal it. Also, there could be a Pokémon professor - type character who gives advice on how to handle Darkrai. For example, they might know some special techniques to keep Darkrai calm while captured.