Is an Owner in pet play the same as a Master?

No, despite the overlapping vocabulary. Pet play's Owner or Handler is a role inside a specific kind of play: training, care, and an animal dynamic. Master or Mistress describes ongoing authority over a person's life.

Pet play is usually lighter and funnier than outsiders expect, with a big gear culture and its own socials. Master/slave is the most formally negotiated corner of this whole vocabulary.

The words collide because English ran out. Ask which someone means rather than assuming.