Out of context, "uniquely American" doesn't mean anything. To say that John Cheever's writing is "uniquely American" doesn't really mean anything. Is his work solely American and completely untouched by any other influence? Is his work different to all other American work and therefore unique? Perhaps I'm being pedantic and becoming too preoccupied with the literal meaning of the words to understand the sentiment of them, but I can't agree or disagree that Cheever's writing is "uniquely American". Call it typically, stereotypically or even very American and we could debate it, but honestly, to me, "uniquely American" is nonsense. Just empty words.
Despite this, I think being identified with a particular culture can be very effective. If you're writing a non-fiction piece, or a fiction piece intended to highlight a particular aspect of a culture, then capturing the essence of said culture is crucial. I just don't think that Cheever's writing is "uniquely American". It sounds like the kind of thing someone puts in an obituary, when they don't know the person but have to say something nice about them.