Brad is a hyper-human name on a pet — a perfectly ordinary American man's name from the mid-20th century attached to an animal, which creates an immediate double-take and then a slow-building affection. The humor is that Brad is exactly the kind of name that nobody would invent for a pet, which makes it ideal for a pet.
The Deadpan Human-Name-for-Pet Aesthetic
Brad belongs to the growing genre of pets given thoroughly unremarkable human names: Gary, Kevin, Brad. The joke works because these names carry absolutely no animal associations — they signal a specific middle-management energy that is deeply funny on a dog who has never filed a quarterly report. This is an owner-type signal as much as a name choice.
The Brad Pitt Factor
Brad Pitt's cultural omnipresence means the name isn't entirely without glamour, but the pet-naming context almost certainly leans on the generic-human-name angle rather than the celebrity reference. A dog named Brad is funnier precisely because it could refer to any Brad. The human name Brad peaked in the US in the 1970s-80s, which adds a generational comedy layer.
The Counter-Reading: The Joke Has a Ceiling
Human names for pets are very funny for the first year and then just the pet's name. Make sure you love the name itself, not just the initial reaction it generates. Gary and Kevin are close cousins in this aesthetic if Brad feels slightly too polished.