I do identify myself by state if people ask. When I open my mouth, it's pretty clear I'm from the US, so I usually just say New York. (And then get a lot of interesting takes on the NY accent..). Like I said earlier, I really only have had that conversation once here when someone asked what my background/what my parents were. I feel like it just doesn't come up here a lot.
But I never just say American, because that's usually pretty obvious. I'm a New Yorker. I'll be a walking/talking guidebook for anyone who is visiting. I don't know (or care, really) if people find my pride in my city and state to be silly, but denying or playing down that part of me would be odd.