As far as I understand, in a practical sense "red prince" is just a trait unique to the red prince character (who also has other more generic traits). So it sounds like you're really just talking about traits.
Did you mean you want to write your own flavour/background text for your custom character? I don't see any reason why they wouldn't let you do this. It should be easy, but it also doesn't seem very important.
Or are you hoping for something like XCOM 2 or the Sims, where you can build your own character with a backstory and upload him to the Steam Workshop?
Ok I'll try to explain further.
YES "The Red Prince" gets the "Red Prince" tag which is unique to that background. Yet he also gets unique quests, he uncovers aspects of his past, and has NPCs specifically respond to his background.
And YES all the backgrounds get 2 additional tags (each race also gets a tag). These tags only effect conversations so far as giving you a few additional options.
However the "Generic" background will never get unique quests, NPCs, locations, or intrigue.
And the "Generic" Background is the only background that allows you to make your own character.
