Is the owner of a dog called only "master"? Isn't "owner" also interchangeable?
gz15)Every afternoon, a dog named Hachi went to train station in Tokyo to meet his master's train. One day, Hachi's master died while at work.
If the dog's owner is a woman, she's his mistress. You could use 'owner' but if you know their gender 'master' or 'mistress' makes it sound a bit more personal.