Snowie is the affectionate diminutive of Snowy, a name applied to white-coated animals since at least the days of Tintin's famous white terrier (called Snowy in English translations). The IE ending makes it feel like a household nickname that hardened into a proper name through daily repetition. For a white or light-coated female pet, it's warm and entirely satisfying.
The Tintin Connection
Snowy is one of the most famous fictional dogs in the world: Hergé's loyal, brave, white Wire Fox Terrier who accompanies Tintin on every adventure since 1929. Owners who choose Snowie are often drawing on that reference. It implies loyalty, intelligence, and a literary quality. Westies and Bichons wear it most naturally.
The Coat-Color Logic
Snowie operates at the white end of the coat-color naming spectrum. It works best when the dog actually is white. On a Samoyed or Great Pyrenees, it's both description and name simultaneously.
The Counter-Reading: Very Literal
Snowie tells you the dog is white and that the owner found that the most defining thing about it. Some owners find that purity of intent exactly right; others find it too simple to sustain over years of daily use.