Bacchus is the Roman name for Dionysus: the god of wine, festivity, and productive chaos. As a pet name it's an explicit statement: this animal is a source of joy and occasional disorder, and we are naming it after the deity most associated with exactly those qualities.
The Wine God Reference
The cult of Bacchus was famously raucous. A dog named Bacchus is expected to honor that tradition: enthusiastic at greetings, unimpressed by rules, fundamentally interested in pleasure. Owners who choose the name are usually laughing about their dog's personality. Labrador Retrievers with their constitutional enthusiasm for food are natural fits.
The Classical Mythology Naming Wave
Mythology names on pets: Zeus, Apollo, Artemis. Bacchus is less common than Zeus, which gives it distinction. The name's human usage is effectively nonexistent, making it entirely available as a pet name without confusion.
The Counter-Reading: Commits Hard to One Personality
Naming a dog after the god of excess creates an expectation. A calm, dignified dog named Bacchus produces permanent irony; an actually chaotic dog produces something closer to inevitability. Both outcomes are fine, but owners should know which one they're signing up for.