Ulfric is borrowing very heavily from Nordic legend for his war campaign. He challenges High King Torygg to a duel in the "old way" for leadership, and uses the Empire's ban on Talos worship to further his cause. How would it look politically if he refused to allow the Dragonborn, a hero of Nordic legend, to live in his city?
Regardless of what race the Dragonborn is, refusing him entry into his city would be highly damaging. He's better off granting him the same rights as any Nord (including purchase of property), since history will probably not remember anything more than his title anyway.