Bill appears 64 times at rank 1604 on male pets. This is a mid-century American given name applied to a dog by an owner who either has a personal association with the name, has a fondness for giving dogs old-fashioned human names, or named the dog after a specific Bill. Given the name's prevalence in American culture, that could mean almost anyone.
The Generic Human Name Register
Bill peaked as a baby name in the 1940s and 1950s, making it warmly familiar to multiple generations. On a dog, it functions the way Bob or Dave does: a plainly human name that refuses to perform anything special, and that plainness is exactly the point. It belongs with Larry and Gary in the mid-century male name set occasionally applied to pets for deadpan effect. The human name comparison is at /names/bill.
The Named-After Pipeline
Many dogs named Bill are named after a specific person: a grandfather, a neighbor, an uncle whose personality the dog shares. A dog named after your grandfather Bill carries that relationship forward in a way names like Biscuit or Ranger cannot. Beagles and Basset Hounds carry the old-fashioned plainness of the name well.
The Counter-Reading
Bill on a dog provokes mild amusement from strangers every time. The name is so plainly human that the gap between form and subject is immediate. Owners who chose it almost always find this funny on purpose.