No info exists on the specific origin of this hybrid; though a Pug-Dachshund mix has probably existed naturally for a long time, no breeder or breed club has claimed its creation. While it's obviously unclear when the first Daug came to be, the breed likely gained popularity during the designer dog "explosion" in North America in the late 20th century. There is, however, plenty of info about the history of the parent breeds.
DACHSHUND: Originating in Germany as early as the 15th century, the Dachshund was developed to hunt badgers (the German dachs = "badger," and hund = "dog") and other burrowing game. Over the years, the original smooth-coated variety, and its long body and short legs, were refined by crossing the Braque (a small French hunting breed) with various terrier breeds; the terrier DNA eventually produced the wire-haired variant as well. The long-haired Dachshund came about by adding assorted spaniel types to the mix. Today the Dachshund is more prized pet than hunter, and ranks 12th on the AKC's list of 201 recognized breeds.
PUG: This breed originated in China around 200 B.C., and is believed to be the smallest variant of the ancient Molosser types. After being a favorite pet amongst Chinese royalty for centuries, the breed's popularity began to spread throughout Europe in the 15th century, when traders brought the little dogs back with them after expeditions to China; the breed became especially popular in England in the 1800s, and it began appearing in the U.S. as well after the American Civil War. As of 2021 the Pug is 28th out of 201 recognized AKC breeds.