Hot dog is a type of sausage. My husband calls it sausage and I correct him, or ask him to clarify, even though it is a sausage. If he tells me to buy sausage at the supermarket, I need to know what kind.
Americans call that type of sausage a hot dog. I grew up in New York calling it a frankfurter.
All other sausages are called sausage and sometimes further defined as well: bratwurst (brats), Italian sausage, kielbasa, etc.