Is a hotdog a sandwich?

Hotdogs have been sandwiches since their inception

Question: Is a hotdog a sandwich?
Position1 of 2›: Yes, a hotdog is a sandwich
Argument‹2 of 2: Hotdogs have been sandwiches since their inception

In the US, hotdogs were first referred to as sandwiches. Since then, the hotdog has become a quintessential part of American food culture and its status as a sandwich must reflect that.

The argument

Hot dogs were first introduced to America in the 1800's, in New York's Coney Island. As one of their first nicknames, they were referred to as both "Coney Island Sandwiches" and the "Frankfurter Sandwiches". Perhaps this does not apply to the original creators of the food item, the Germans, but it certainly applies to America. The hot dog was called a sandwich in its original glory days, and thus still must fall under its original category. It doesn't make sense to change the categorization of a product just because people are stubborn in opinion.

Premises

[P1]When hot dogs were introduced to America at Coney Island in the 1800s, they were nicknamed "Coney Island Sandwiches" and "Frankfurter Sandwiches." [P2] The food was therefore classified as a sandwich from its earliest American days. [P3] It makes no sense to change a product's original categorization simply because opinions later became stubborn. [C] Therefore, because hotdogs have been sandwiches since their inception, a hotdog is a sandwich.

Counter-arguments

As mentioned, the hot dog is not considered a sandwich EVERYWHERE. If it's not a universal thing, why should every single hot dog be categorized as a hot dog? I f everyone can't agree on it, why place a controversial label upon it?

Rejecting the premises

[Rejecting P1] Early nicknames record how an unfamiliar food was sold to American customers, not a settled classification; they sat alongside other names for the same item. [Rejecting P2] A nickname is not a taxonomy. That vendors called it a sandwich no more establishes the category than "hot dog" establishes what it is made of. [Rejecting P3] Ordinary-language categories are fixed by current usage, not by first usage, and the argument gives no reason why the earliest label should bind later speakers — particularly when, as the counter-argument notes, usage is not uniform even now.