While I appreciate the topological approach, I hold to a linguistic and practical reason for a hot dog, as it is usually eaten, not (typically) being a sandwich:
what does it mean for a thing to be “sandwiched”? It means pressed on two sides, held together by the force of that pressure.
what is the difference between a hot dog and a hero/po-boy/sub? Well, heros and po-boys are held together by the bread. You can turn them on their side, and they should not fall apart, because the primary force holding them together is pressure on either side of the bread. Hot dogs, at least in my limited experience, are defined by their toppings, which are placed atop the frankfurter, and held in place by gravity alone.
As such I give my typology: if the primary force holding your dish together is pressure on two sides from a retaining material? Sandwich. If the primary force holding it together is gravity? That bread is being used as a trencher. As such, most hot dogs, most tacos, bread bowls and other such things are all basically just a version of a bread bowl or bread plate. For this reason, I call them “Trenchers”. Pizza is not primarily held together by gravity or by sandwiching forces, and thus is normally neither of these. Pizza’s primary force maintaining its integrity is the cheese and other sticky things holding onto any toppings. As such, pizza would be equivalent to toast with spread, cheese, and other toppings, similar to garlic bread. All functionally just “adorned breads”.
So, to reiterate, I don’t disagree that hot dogs can be sandwiches, but in general practice, I do not believe they qualify, much like most tacos do not qualify.
