Yes, the phrase “hot dog” does tie in to the Dachshund breed.
In 1852, frankfurters (hot dogs) were ‘born’ in Frankfurt, Germany. The better known frankfurter was made by a butcher known for his Dachshund. His frankfurters became known as Dachshund sausages.
Around 1895, sausage vendors sold Dachshund sausages outside student dormitories, and their carts became known as dog carts.
The first record use of the term hot dog occurs in an 1895 Yale Record, although the name didn’t catch on until 1902, when vendors hawking Dachshund sausages at a cold Giants baseball game urged patrons to get their “Dachshund sausages while they’re red hot!”
A sports cartoonist, Tad Dorgan, was nearing deadline and quickly drew a cartoon of a frankfurter with a tail, legs, and head so it looked like a Dachshund. Not knowing how to spell the name, he captioned it “hot dog!” The name stuck.
The frankfurter may gave gotten its hot dog nickname from the Dachshund, but the Dachshund got its wiener dog nickname from the frankfurter.
These kind of facts, and more, can be found here:
Barron’s ‘DACHSHUNDS’ book
Dachshunds – The Owner’s Guide From Puppy To Old Age