Cubby is a name built around smallness and softness. The word conjures a cub, a little hiding spot, something tucked away and round. On a puppy or kitten it arrives naturally in the first week, when the animal is still discovering corners of the apartment and fitting into spaces it will shortly outgrow.
The Baby-Stage Name Problem
Cubby is one of those names that fits perfectly at eight weeks and requires commitment to maintain at three years. A large Great Pyrenees named Cubby is either deeply ironic or a testament to owner loyalty. On smaller breeds, Pomeranians and French Bulldogs, the name scales better because the animal never leaves the round-and-dense phase that inspired it.
Sound and Warmth
CUB-by is two syllables with a soft B and a short U: phonetically warm, easy to say with affection. It doesn't carry any edge or formality, which is either its strength or limitation depending on whether you want a name that commands attention at the dog park.
Counter-Reading: Chicago Cubs Association
In Chicago, Cubby has an unavoidable baseball association. Nationally, it reads as straightforwardly cute. Compare with Bear for the same cub-adjacent warmth with more universal recognition and no regional asterisk.