Is it acceptable to call a hot dog a sausage?
Solution 1:
In short, yes, it is acceptable in AmE to call a hotdog a sausage, but it does seem strange when you put it like that because ones idea of sausage in AmE is nothing like a hotdog.
Technically a hot dog (or frankfurter) is a specific type of sausage. There's no way around that (processed meat shaped like a tube). The kind of meat, casing preparation, color, taste, etc all distinguish the kinds of sausage.
In most people's minds, however, the canonical sausage is not a frankfurter. In the US, the hot dog is very common and most other sausages are not. Also, most non-frankfurter sausage in the US is presented in a form that is very much unlike the traditional frankfurter, usually sliced in circular shapes (luncheon meats like bologna, mortadella, salami, and in a different setting breakfast patties), or ground as in ground sausage for mixing in other things.
I would go so far as to say that, while the frankfurter is in the traditional 'wurst'-like appearance, an AmE speaker would probably also deny that a brat (a bratwurst) is a kind of sausage. Or would even deny, despite its name, that a 'Vienna sausage' is a sausage.
A very different example, 'shoe' and 'sandal'. Technically a sandal is a type of footwear, and one might consider 'shoe' as a synonym of 'footwear'. But in practice, if someone says, 'put some shoes on', and you put on sandals, the first person might say, 'no you didn't understand, put some -real- shoes on'.
Which is all to say that, what you call something is not necessarily an accurate reflection of a technical relation. Even though 'sausage' is technically a hypernym (the name of the concept that includes the concept of another word) for hotdog, the canonical example of (and general usage of the term) sausage in AmE is definitely not a frankfurter.
Solution 2:
All squares are rectangles, but when you say "rectangle," the image that comes to mind for most people is probably a rectangle with unequal length and width. The expectation that comes from the word rectangle is a set of objects that's smaller than the set that comes from the definition of the word.
I think it's the same with sausage. It's true that hot dogs, pepperoni, salami, balogna, kielbasa, and chorizo are all sausages. But if you meant one of those kinds of sausages, you'd normally use its more specific name. The word sausage then takes on an additional meaning that's something like the things that qualify as sausages for which we don't know a more specific name. If your spouse asks you to pick up a package of sausages at the store, you might ask: "Hot or mild?" or "Fennel or sage?" or "Turkey or pork?" but you wouldn't ask "Hot dogs or pepperoni?" because the answer is obviously neither of those.
Solution 3:
I am Australian but spent a few years in the US as a child.
While I can agree that a hot-dog is technically a type of sausage, I would never call a hot-dog a sausage or a sausage a hot-dog, and I believe that this is a widespread predilection.
A hot-dog (or frankfurter) has a red skin and is already cooked, and merely needs to be heated in boiling water before being consumed. It can also be baked, of course, as when added to macaroni and cheese.
A sausage can be cured (eaten cold, except when added as a topping to pizza) or raw (eaten hot after frying or grilling).
Never the twain shall meet, or be confused with each other!
Solution 4:
In the UK, there is a difference, but it is a different difference. A Hot Dog here would normally refer to a sausage in a roll, not the sausage on its own.
A sausage would normally refer to a skin filled with meat and who knows what else. Eating it raw would be akin to licking a pig, and not recommended, but cooked (normally fried or grilled) are fine, if you like meat. As a veggie, I prefer the sausages that do not involve meat at all, and so are not really sausages.
And there are also pre-cooked or smoked sausages, which are also sausage - often in the singular - eaten in all sorts of ways. Normally, these would be referred to by specific names, and tend to come from Germany or Italy.