Cod, I think your confusing National identity with Cultural identity.
All Americans are 100% Americans no matter how long or they've been there, or freshly arrived they are. As an American, when I lived there, I assumed everyone was American, no matter what accent, language they spoke, color of there skin was, etc - unless they said otherwise. That's the National identity.
All Americans have a Cultural identity too. When they say they are Italian American, they are not referring to the passport their ancestor carried around, they are referring to the traditions, the language, the food, the family attitudes, etc.
Because of this, it quite easy for Americans to be whatever / whatever Americans. This is far more ambiguous in Europe as nationality and culture are blurred together ... if someone says they are are British, you'd assume that's the passport they are holding, but if they said that with a French accent, they'd probably get a funny look.
Regarding when people become naturalized Americans, they are giving up their previous National Identity, not their Cultural one.
Hope that clears it up!