Some cards, like the Oni of Souls, Form of the Demon, or Form of the Beast, refer to the player “having” Oni or Monsters. For this to take effect the figure in question doesn’t need to be on the board, the player simply must own it. For cards that refer to Strongholds (Form of the Kitsune), they must be on the board, since you don’t own the Strongholds in your reserve.