Samuel on a pet occupies the same category as Elizabeth and William: a full-weight human name applied to an animal, where the humor and the affection operate simultaneously. It's three syllables of Old Testament gravity on whatever creature has decided to eat the corner of your couch, and that combination is the whole point.
Biblical Weight, Domestic Reality
Samuel derives from the Hebrew Shmu'el, traditionally interpreted as "name of God" or "God has heard." It's one of the oldest continuously used names in the Western tradition, a top-10 US baby name for most of the past two decades, and it now appears on dogs and cats with approximately equal dignity. The human name Samuel currently ranks in the US top 25 — high enough that a pet Samuel is operating in the same cultural moment as human Samuels.
Nickname Architecture
Sam is the obvious daily-use form, and it functions beautifully as a pet name on its own. The full Samuel on paperwork with Sam in practice gives owners the best of both registers. Sam independently ranks well in pet registries, which suggests many registered Samuels probably live as Sam. Labradors and Golden Retrievers are natural Samuel carriers.
The Formality Spectrum
Samuel sits at the serious end of the over-naming spectrum — more deliberate than, say, Charlie, less theatrical than Lorenzo. It says: this animal is a full member of the household, deserving a name that treats them accordingly.