If people assume it's "just a type of sausage" it suggests a dictionary entry is needed to explain otherwise.

It's a term referring to a small set of types of sausages served in a specific small set of ways. In some places, a hot dog can be used as a synonym for the predominant type of sausage most common in hot dogs in that place, but the term is still more commonly referring to the assembly of a wiener or frankfurter wrapped in a bread of some sort.

A dictionary is an enumeration of words. A thesaurus is a mapping between existing words.

Every word in a thesaurus belongs in a dictionary.

All words in a thesaurus would generally also be in a dictionary? The difference between a thesaurus and a dictionary is what each tells you about a word.

It’s a type of sausage, but they are definitely not synonymous. At least not in American English.

In the US, if you ordered a hot dog and got a sausage (or vice versa), it would be very reasonable to return the item and ask for something else. They are culturally completely different, the same way Cheerios in milk is not another cold soup like gazpacho is.