A sandwich to me is something served betwixt two slices of bread! And then it is described by its contents. Wafer thin rare beef, horseradish sauce, shredded lettuce on hand sliced granary bread. Perfect.
Anything else is described by the type of roll used - baguette, cob, bap, ciabatta, panini, cheese topped roll etc etc.
Where I have seen anything described as a sub here it tends to be long and white like a hot dog roll - though with a softer top and the same width all the way down - usually dusted with maize.
We have a place called Po'Boys, but it sells the same mix as everyone else. Although I go there sometimes and have my own creation of Stilton, hot bacon and jalapeno peppers in a French stick, because it's the only place with all the ingredients.
Our sandwiches aren't as exciting as yours
My current favourite at my local deli is even called a New Yorker. Sigh. But at least it's on a proper Jewish style bagel. We now get so-called "New York style" bagels in the supermarkets. Not sure if they are anything like the bagels in New York (didn't have one while I was there) but they certainly aren't anything like the bagels I've eaten from Jewish bakeries or down Brick Lane in London.