Brian appears 63 times at rank 1622 on male pets. It's a perfectly ordinary human name that reads as mildly absurd on a dog, which is precisely the aesthetic a certain type of owner is going for. The humor in naming a dog Brian is that the name is so resolutely human that it refuses to cosplay as a pet name.
The Deadpan Human-Name Strategy
Brian joins a category of pet names that work through understatement: Dave, Jeff, Gary. The joke is the mismatch between the bureaucratic normalcy of the name and the animal it belongs to. Brian Griffin from Family Guy is the most famous dog to carry it, and probably amplified the choice for a generation of millennial dog owners who grew up with the show.
Sound and Breed Fit
BRY-en is two syllables, clear and unambiguous. The Brian Griffin connection makes Labrador Retrievers the obvious breed choice, but the deadpan-name aesthetic works on any breed. The human name is at /names/brian.
The Counter-Reading
Brian is not trying to be a pet name. That's the whole point. Owners who want something that sounds like a pet name should look elsewhere. Owners who find the incongruity genuinely funny, and are prepared to explain it approximately once a day, will be very happy with Brian.