Dr ranks #3379 with 24 registered male pets. It is, as far as the dataset is concerned, simply the honorific — two letters, no period in the registration data — applied to a dog or cat as a name. It's the kind of choice that requires no explanation to anyone who hears it and immediately makes the animal seem 40% more distinguished.
The honorific as pet name
Using titles and honorifics as pet names is a small but consistent tradition. Dr, Professor, Major, General, Captain — all appear in pet name data with some regularity. The appeal is obvious: the contrast between the formality of the title and the reality of an animal that eats off the floor is inherently comedic. A cat named Dr carries an air of unearned authority. A dog named Dr seems to be judging your choices from a position of professional expertise. The joke writes itself and never gets old.
Usually followed by something
In practice, most pets registered as "Dr" are presumably "Dr Something" — Dr Pepper, Dr Doom, Dr Woof — where only the first word made it into the registration field. The standalone Dr is either an incomplete registration or a deliberate minimalist choice. Either way, it functions as a naming philosophy: maximum dignity, minimum syllables. Poodles and cats seem to receive this honorific at higher rates than other animals, possibly because both species have a natural air of intellectual superiority that the title reinforces.
The naming tradition it belongs to
Dr fits alongside other title-names in the dataset as part of a broader trend of giving pets names that acknowledge their evident sense of self-importance. Professor and Captain occupy the same space. If your pet clearly believes it is in charge of the household and deserves professional recognition for managing your affairs, Dr is the appropriate documentation of that arrangement.