Frankfurters, franks, wieners, weenies, dogs are all names for hot dogs. Technically, there are differences in their ingredients, but this is mostly ignored with labeling describing the ingredients. For example, franks were made with beef and the name was to signify the difference to hot dogs that were made with pork.
Not using an article with the word "sausage" in English turns it into a non-count noun, and refers to ground sausage in either its raw or cooked form. But the Ukrainian сосиска is very much a count noun and refers to individual sausages (or hot dogs). So yes, the article does matter.
This is where the numerous "rules" become useless: it's easier just to become accustomed to what people say. The rules of articles have limited logic (and are often different in other languages that use them e.g. French La France but English France).
English learners find a word like "horse" in a dictionary and try to use it directly. Native English speakers never heard the word "horse" alone: it's always a horse or the horse.