- You're an odd reptiloid no one has seen before, but happens to look exactly like that cool Dragonborn on that long-forgotten roleplaying game cover you found on the flea market.
- You're a cybernetics/magic/genetic/zeno experiment gone wrong/right.
- You fell through a portal from a world where your kind exists.
- You're member of a cult that worships fire or the god of fire, and that's why you can breate fire and use Dragonborn stats.
- You're a changling that assumed this form based on a description from a book of made-up animals, but got hit by lightning and can't end it now.
- You drank an unlabeled but probably spoiled potion and that's the result.
What the compromise could look exactly will depend on what makes sense to DM and the player (and possibly others in the group) in question. There is no universal answer. But there is much more likely to be an answer than it is that the DM's campaign and the player's character idea are 100 % incompatible and unmoveable.
If you want to add dragonborn or any other race when you DM, go for it. I personally don't care for that kind of world building, which is why I tell people when I invite them what my restrictions are.
If that means I'm not the DM for you, so be it, I have no problem recruiting or retaining players.