Presumably, he's the "First Avenger," rather than the "first Avenger." Think analogous to Namor being the "First Mutant." As in, they were This before everybody was This.
(Not that movies should be at all beholden to what has happened in the comics. They're not courting us as customers, because, with modern sales numbers, every existing Captain America fan buying a 3D ticket to watch it would net them the princely sum of...about three million bucks. That probably doesn't even cover craft services.)