The lore says that not all Dragonborn are related to Nords or the Imperial line, but rather appear and disapear in various locations throughout history. And it's more of a power or talent than what's in your blood. One of the main roles of the Greybeards is to track these occurences and unravel the circumstances surrounding them. :spotted owl:
So no, any race being Dragonborn fits the lore perfectly.
Listen to this man...bunny....thing. He has a owl, and he is correct.
I never thought this in Morrowind or Oblivion so I don't see why I would in Skyrim.
In my own personal "cannon" I consider the Nerevarine to be an Argonian. It just seems fitting. Especially when you consider it is the Nerevarine who strikes down the tribunal and puts Helseth on top. (Helseth later abolishes slavery) Not to mention it's the Nerevarine's actions that lead to the fall of Morrowind and its destruction. Though I really blame Vivec and Nerevar for that.
And in turn I'm fairly certain I'm going to make my "cannon" Dovahkiin a Dunmer. That fits for me because he will be coming from a broken people and it will be kind of a repayment for the Nords allowing the Dunmer (their ancient enemies) to come settle and build a city in Skyrim (one of the foreign mags said that. Don't remember which)
As for the Champion of Cyrodiil, the Hero of Daggerfall and the Eternal Champion I haven't really decided my own "cannon" for them. No race really "fits" better than another for them in my opinion.