Handsome appears 64 times at rank 1609 on male pets. This is a direct compliment turned proper name: the owner looked at the dog, found him beautiful, said "you handsome thing" enough times that the word became the name. It's a registry artifact of affection at its most transparent.
The Compliment-to-Name Pipeline
Handsome belongs with Gorgeous and Beautiful in the category of compliments that became registered names through repetition. Most owners called the dog Handsome as a term of address rather than a chosen name, and it stuck before a proper name was decided. The name is most common on male dogs whose physical appearance genuinely startled the owner: a particularly striking Husky, an unusually beautiful Weimaraner, a dog whose looks warranted commentary from everyone.
Breed Fit
Weimaraners, Vizslas, and Siberian Huskies are the most common recipients: breeds that reliably prompt strangers to stop and comment. A Handsome who is genuinely handsome is a name that functions as a daily observable fact. The name is emphatically male-coded and appears almost exclusively on male dogs.
The Counter-Reading
Handsome requires the dog to live up to it, and most dogs so named apparently do. The only liability is that it's a description rather than a name, and descriptions can become slightly awkward when the dog ages. A twelve-year-old Handsome is still Handsome, and that's the point.