yes, we do!
no so much here is australia, as we have many similar foods, but we might refer to eating at US style restaurant (something like Hogs Breath = chain of "steak" houses, or buffet type places like sizzler) as 'going for american food', we say 'american hotdogs', meaning low quality sausage, on low quality soft white bread roll with tomato sauce and.. american mustard!
and places will say they serve 'NY or Boston style pizza' or 'NY bagel' or 'texan bbq' etc or say 'american hamburger/cheeseburger' to differentiate from Oz style burgers, which by law must have on them salad including beetroot. (that is a joke. of course it is not a law).
other countries do have cuisine movements based on 'american' style food, that they do call 'american', but yeah, it's not quite in the same way we say we're eating 'chinese', 'thai', 'polish' etc about cuisines which are more distinctive...
I'm not american, and don't live in USA (shock! horror!)