In most of the English-speaking world, there is no possibility of genuine ambiguity about what the meat in "hot dog soup" consists of—beyond the fact that hot dogs themselves are sometimes characterized as "mystery meat." Objectively, "hot dog soup" is no more ambiguous than "hot dog on a bun," and no one familiar with the U.S. term hot dog doubts what that is.
On the other hand, hot dog is a bit of an oddity in that Merriam-Webster spells the noun (referring to the sausage) as two words and the verb (meaning to show off by doing something flashy rather than merely workmanlike and efficient) as one word, closed up. A Google Ngram chart for the years 1900 to 2005 matching hot dog (blue line) against hotdog (red line)—with no differentiation between how the two terms are used—looks like this:
A specific comparison of plural noun forms—hot dogs (blue line) versus hotdogs (red line) across the same period yields a rather similar-looking chart:
It wouldn't surprise me if the closed-up hotdog eventually overtook the open hot dog in all senses of the term, but that obviously hasn't happened yet. If I were in charge of producing the printed menu for a restaurant that offered hot dog/hot-dog/hotdog soup, I might be inclined to run hotdog as one word, closed up, just to minimize the hilarity of easily amused diners. But then again, maybe a little hilarity in that setting is a good thing—as with the menu at a Chinese restaurant that offers such choices as "Strange flavor chicken."