A non-nordic Dovahkiin I find weird.
Not because of the whole "I'm a Dovahkiin!" issue, but because the game was obviously built and centered for a Nordic player, and assumes that's the only race the player will choose. There's no additional proving required to, say, Ulfric if you happen to be an Altmer Dovahkiin. Or the Blades (who are wary of Thalmor plants to begin with. And the message they leave has too many "if's" attached to it, in my mind.) The only real indication for your race is combat taunts and random lines from NPCs, and I don't really count those, since they amount to throaway lines.
Morrowind was centered for a Dunmer player, but they wrote it well enough that you had to look hard for it, as the circumstances surrounding the main quest were equally applicable to any race/gender combination. And Oblivion... really lacked any indication of what race they had in mind when designing it, other than (likely) male.