The "big chain" type certainly exists basically everywhere by now. There is a reason why the Big Mac Index is suitable as an economic indicator: you can calculate it for almost all countries in the world, because you can buy a Big Mac in almost all countries in the world.
A second type of "American restaurant" is much harder to find. It is the kind of small diner which serves grilled cheese sandwiches, thick pancakes and other American style food, without being a chain. I have seen this in places with large expat populations, but most Europeans will probably spend their lives without ever having been in one. I can't talk about other continents.
A third category of "American food" would be American homemade food. Chicken pot pie, eggplant parmesan, Southern biscuits, that kind of stuff. I have not seen it served in any restaurant in Europe. I have never seen an "American home food restaurant", nor an "American fine dining restaurant". They could exist, but as I've visited many large European cities and lived in places with a large number of American expats, they are likely to be quite rare, or maybe clustered somewhere I haven't been.
What I have seen more in later years seem to be fancy burger places - they are sometimes chains, sometimes not, but I would distinguish them from fast food chains because they tend to have things like mushroom-and-arugula burgers on sourdough bread. Also, you nowadays see more burgers in restaurants which are some sort of middle tier without special relation to a specific cuisine - more of a nonfrench equivalent of a bistro. Also, Starbucks style chains (including Starbucks itself) are pretty established - while Europe had an extensive coffee culture before them, the type of drinks served there is pretty distinctive.
American packaged food such as marshmallows or pumpkin pie filling can be also found in specialized grocery stores selling American products, and as a seasonal article in large European supermarket chains like Lidl, who tend to have "American week", "Greek week" etc. in rotation. A few selected American products are also found as staples in most stores, such as Snickers bars or Coca-Cola, or are less available but still within easy reach, such as Jelly Belly candy.