During any run, collecting the Blue Demon Aspect Stone will swap your own Aspect Stone and changes the character's appearance to look like the Blue Demon. This Aspect Stone replaces whatever character stats you had previously. As an example, Playing as Eye and grabbing a Blue Demon Aspect Stone will increase your health from 1 to 3 and changes your character's ability. Swapping Aspect Stones does not affect Organs.
The Blue Demon is not a permanent character in the Library. It can only be used inside the Deathstate. Because of this, there is also no ending for this character either.