The player must be dragonborn from birth because of the nature of what that means. People who are dragonborn are born with aedric dragon souls rather than normal black souls. That isn't something that changes later in life, as your soul is the essence of who you are. You are actually a dragon, just living in a humanoid form.
Note that you technically
didn't exist before appearing on that carriage. You didn't have a history or anything until you awoke, and then you suddenly did.
Beyond that, we also don't know what kind of soul manipulations are possible. Soul transference is possible, so maybe your original soul was pushed out and a dragon soul was put in (maybe you did die in the ambush, and a dragon soul made its way in to bring you back to life). Or perhaps the gods changed your soul into that of a dragon's, when it wasn't before. Or perhaps by sheer determination you changed yourself into a dragon. Or maybe Alduin accidentally changed you when he hit you with that shout on the chopping block.
It's a bit heavy-handed to say you must be born with the soul of a dragon to be a dragonborn, especially when an in-game book questions whether it's a trait people have from birth, or if it's gifted to them at some point in their life.
Very few realize that being Dragonborn is not a simple matter of heredity - being the blessing of Akatosh Himself, it is beyond our understanding exactly how and why it is bestowed. Those who become Emperor and light the Dragonfires are surely Dragonborn - the proof is in the wearing of the Amulet and the lighting of the Fires. But were they Dragonborn and thus able to do these things - or was the doing the sign of the blessing of Akatosh descending upon them? All that we can say is that it is both, and neither - a divine mystery.
- http://imperial-library.info/content/book-dragonborn