A burger is an interesting example, as some formulations would presumably trigger the mandatory origin labelling requirements applicable for beef products.
Ignoring that element, calling a product an "American Burger", "Italian Pasta", "German Beer" etc is almost certainly going to be seen as implying the product is of that origin.
There are some exceptions where there is a defined legal name - e.g. London Gin doesn't imply that this is gin produced in London
Similarly there would be some cases where a customary name that includes a geographic reference is reasonable and acceptable - e.g. a Frankfurter sausage would likely be generally accepted as a type of sausage, rather than one that is necessarily from Frankfurt.
I think that you'd have to go with "American style" in your example. The overall context of the packaging would potentially need to be considered in terms of whether this would trigger other origin labelling obligations - if it is very prominent and plastered with American flags, then you may need to clearly state the real country of origin (and potentially the place of provenance under Article 2(2) of 1169/2011) so as to not risk misleading the consumer.
The regulations are necessarily quite broad and therefore they don't specifically define every potential acceptable/unacceptable case, and that puts the onus on the Food Business Operator to ensure that they're compliant, and in this type of scenario that is primarily going to be about not misleading the consumer.
This is therefore all my personal opinion only, but for what it's worth I would not call a product an "American Burger" unless it was made in the USA using beef from the USA.
FoodDrinkEurope has a guidance document linked from their website here that might be a useful reference for you: https://www.fooddrin...ary-ingredient/