I totally see Krypto as something Gunn would be drawn to. I'm just wondering how he got there. I mean I can't see Clark dropping him off for grooming as I assume he would break their shears? I saw a theory on Reddit that this Krypto is the product of animal testing for the suspected Superman clone, which make a lot of sense.
Maybe Lois, Supes, and/or Terrific found a dog at a LexCorp testing facility while investigating the frame-up. They drop it off at a shelter for adoption or something. After Terrific realizes the Man in Black is a Supes clone, he figures the dog has powers and goes to get him because he needs his powers to rescue Superman. Hence the difficulty he seems to be having convincing Krypto to fly or something and his desperate use of dog treats. It's all new for the dog.
And to be honest, Krypto being a rescue pet seems way more Clark than some magical Kryptonian dog.