The Dog was a genetically-engineered lifeform created by Ensign D'Vana Tendi in 2380, during her free time aboard the USS Cerritos. This was first spurred on by personnel in the Medical division being encouraged to develop side projects.
Tendi created The Dog from inert carbon, hand-edited all six billion sequences that were encoded in her DNA, which included the reorganization of her mitochondria, and her protein bonds successfully accepted the Cas9 snipping without any cellular degradation. The end result made her 5% faster and 6% smarter than a typical dog.
Although she was modeled after a domestic dog from Earth, The Dog had abilities beyond those of normal canines, including the ability to shape-shift, walk on walls, turn her eyes into bat-winged creatures that could fly out of her mouth, speak in a language recognizable by the universal translator, hover, and spit lightning. Tendi deliberately gave The Dog these features, as her homeworld of Orion did not have dogs, and Tendi mistakenly believed these were attributes of "normal" dogs.
As a "freak" that defied categorization, Doctor T'Ana referred to The Dog as "Mr. Whatever The Hell That Thing Is". The Dog was taken by Division 14 to Endicronimas V, where she could be cared for by the staff of the medical spa known as The Farm. (LD: "Much Ado About Boimler")
Later, Tendi retold the story of creating The Dog to Ensign Sam Rutherford after the loss of his cybernetic implant caused him to experience memory loss. (LD: "No Small Parts")
A promotional "Tech Corner!" review of the Dog was posted a week after the episode aired, by Star Trek on CBS All Access, describing her many enhancements, including "new and improved", "every line of DNA code updated", and "5% faster, 6% smarter". It further listed its abilities: "shapeshifts • talks • hovers • spits lightning".