If cold cuts, etc. is served on a long roll (French or Italian bread), in New York it was called a Hero. In California, it's a Sub(marine) sandwich, particularly with the popularity of Subway. When they first opened, they actually had a map of the N.Y. Subway system & tiny subway seats.
Years ago, the "Club" sandwich was popular in NY. It consisted of three layers of toasted bread & various fillings - usually turkey, bacon, lettuce, tomato & mayo - w/ a side of chips.
A hamburger or hot dog might qualify as a sandwich - any filling between two slices of bread - which puts me in mind of a patty or tuna melt. An open-faced sandwich w/ turkey & gravy was another popular sandwich in NY (served at Howard Johnson's etc. around Thanksgiving). Nathan's in Brooklyn, NY served chow meain on a hamburger bun. May sound odd, but it delicious.
To add to the cultural/ethnic list - bahn mi, panini, gyro, falafel, schwarma, mufletta (sp), stromboli. We have some bagel places that will put together any sandwich of hoice. I like nova scotia lox, cream cheese, red onion & tomato.
One of my favorite sandwiches is served on a buttery, flaky croissant - chicken or tuna salad - or whatever you like. Or, a sweet version with nutella or chocolate & strawberries (also called a panini - if it's grilled & served on ciabatta or crusty bread).
Recently, I noticed sub shops coming out with sandwiches on flatbread - probably to give folks a healthier low-carb alternative.
One of my favorite make-at-home sandwiches/subs is a Philly Meatball - with sauteed onions, green peppers & provolone.