It's probably the opposite, really—I'd say an American is more likely to characterize anyone from the island as "British" rather than English, Welsh, Scottish. Think of how many American opinion polls characterize the "British" accent as sexy, when they're probably thinking of Jude Law's accent rather than scouse or glaswegian
Am American and received a very good education here. They never differentiated England, the UK, or Great Britain that I can ever remember. To most of us, they are interchangeable terms. I remember being curious about it and looking it up myself. You’d think as the country we won independence from, and is now our closest ally, we might be taught that information but no.