Keep clicking on the pets between games - the cat, the dog at a minimum. I clicked every animal on the screen between games LOL - literally! It won't happen after just one game - you have to do it between several games - at least I did.
Thanks, EDP, I followed your suggestion and earned the badge. You were right...it took me five rounds of the game to get it. After clicking on the animals, she walked to at least one in each round, and patted or fed the animal.