Curly appears 65 times at rank 1590 on male pets. The name is a direct physical observation: the dog has curly fur, elevated to a proper name by affection and habit. It's one of the oldest descriptive-nickname-to-formal-name traditions in pet naming.
The Physical Descriptor Tradition
Curly belongs to the same category as Patches, Spot, and Fluffy: names that describe what owners see and then live with permanently. For curly-coated dogs, the name is accurate and obvious. Poodles, Labradoodles, Portuguese Water Dogs, and Bichon Frises are primary recipients. The coat makes the name feel inevitable rather than assigned.
The Three Stooges Layer
Curly Howard of the Three Stooges is one of the most recognizable comedy characters in American pop culture. A dog named Curly carries that comedic, bumbling-but-loveable energy whether the owner intends it or not. The association suits dogs with clownish personalities: the dog who runs into walls, who trips over its own ears, who makes people laugh without trying.
The Counter-Reading
Curly ages beautifully on dogs. An elderly curly-coated dog named Curly has a certain dignity in its honesty. The only counter is that it's fundamentally a nickname, and some owners feel formal pets deserve formal names. Curly disagrees, usually loudly.