Lillie is the Victorian spelling of Lily, the flower, the purity, the easy feminine sweetness of it. Female pets at rank 1179 named Lillie tend to belong to owners who prefer the ornate, slightly antique spelling to the more common Lily. That single detail (the double-l and the -ie) marks the name as considered, not default.
The Lily Cluster
Lily, Lilly, Lillie, Lilli: all variations draw from the same pool of floral, feminine appeal. Lillie specifically was fashionable in the late 19th and early 20th centuries, associated with that era's love of botanical names. Lillie Langtry, the Victorian actress and socialite, was one of its famous bearers. On a pet, the Victorian spelling gives the name a gentle period quality that Lily doesn't quite have. It reads as softer, slightly more formal, pleasantly old-fashioned.
Breed Fits
Lillie suits white or cream-colored breeds particularly well for the obvious floral reasons. Bichon Frises, Maltese, and Samoyeds are natural Lillies. That said, any gentle female dog or cat can wear it. The name isn't limited by coat color so much as by personality energy.
Human Name Parallel
The human Lillie is moderately used in American naming, more distinctive than Lily and less formal than Lillian. On a pet, it reads as a genuine name rather than a borrowed one. It's the kind of name a grandparent might suggest and the owner end up keeping, which is a reliable sign of durability.