Common in American English, a dog's owner will be called the dog's master.
Master: one having authority over.
"When I say come, he comes. When I say go, he goes."
Not used with cats. They do not obey the owner as much as a dog would.
Only called master if the dog obeys or is trained to obey it's master. A well trained dog will do tricks for his master. Otherwise, the owner is is just an owner.