A dragonborn with a draconic bloodline color that differs from their own scale color causing cognitive dissonance and personality swings has been a character idea of mine for a long time. Never thought of my character having two heads though, very cool. I think I would play this as the draconic bloodline’s head growing in progressively over several levels.

As a DM, I would have a conversation with the player about how this would work for them, but as long as it mostly affects roleplay and not mechanics I would allow it. Roleplaying two heads has to be difficult though, I would probably run a trial session in case the player realizes it’s a terrible idea and wants to do something else.

If my player wanted this to have mechanical implications, I would probably tell them to just build their character accordingly. You want to be more perceptive because of your two heads? Take the Alert feat at level 4. You want to dual wield more effectively? Take the Dual Wielder feat at level 8. etc.

Alternatively, I could see two players sharing this character. That could be really fun for a one-shot or short campaign. For all intents and purposes, they would act as two separate characters, with two different initiatives, their own resource pools, their own saving throws etc., would share their ability scores, movement in combat and would have to agree to perform athletic and acrobatic maneuvers such as climbing and jumping. Each character could choose their own class progression independently.