trivia note: the silly dog that accompanies child Falin is most likely Dustrag, the stray mix dog that its owner wanted to ger rid of and the Touden's father wasn't happy when the the siblings took him home.
dog!Laios and Dustrag do have similar design and I think this dog's resembling Laios' feeling about how his father's treatment to him and his sister (unwanted, but they try to stay optimistic and having fun in life)