I'm not suggesting that a hot dog or hamburger don't potentially, technically fall into subsets of a broader group called sandwiches.
However, if nobody ever refers to a hot dog as a sandwich, can we really say that it's a sandwich?
People DO refer to Bugattis as cars.
But I've not once ever seen someone order a sandwich and receive a hot dog, nor heard someone pronounce "I think I'd like to have a sandwich for lunch" and then order a hot dog.
I did however work with a guy at our BBQ restaurant, a dude who was of Mexican-American descent, who brought tacos made by his mama for lunch everyday. When he was ready for lunch, he'd say, "Okay, it's time for my sandwich, I'll be back in a few."
So who knows?
I was going to repeat what I think I read a long time ago--that "taco" is Spanish for "sandwich."
But an English-Spanish translation page provides "emparedado" as the proper Spanish noun.
I also was going to say that a hot dog is not a sandwich because it does not use two pieces of bread. But then neither do a lot of subs/heroes/hoagies, and they are sandwiches.
Still, in common usage, hot dogs are hot dogs, hamburgers are hamburgers, and sandwiches are sandwiches.
Speaking of hamburgers, their origin is somewhat in dispute. Part of the problem is whether or not a ground-beef patty between two slices of bread qualifies as a hamburger. For me, it does not. Growing up, I would have called one of those a ground-beef sandwich.
If a hamburger is a ground-beef patty between two halves of a bun, then this is the best claim to have been the first, per the Font of All Wisdom and Knowledge:
Oscar Weber Bilby
The family of Oscar Weber Bilby claim the first-known hamburger on a bun was served on July 4, 1891 on Grandpa Oscar's farm [outside Tulsa]. The bun was a yeast bun. In 1995, Governor Frank Keating proclaimed that the first true hamburger on a bun was created and consumed in Tulsa, Oklahoma in 1891, calling Tulsa, "The Real Birthplace of the Hamburger."
Earlier claims all seem to be about a patty between two slices of bread.