First, "no bigger than" doesn't mean "as big as". It means "as big as, or smaller than - but not bigger."
If the creature is no bigger than a dog, it follows that it's no bigger than a horse. In any case, "(not a horse)" or "(not a mouse)" doesn't make sense with that sentence.
If the creature is the same size as a dog, it's entirely unnecessary to list any other animals that it is the same size as (or indeed that it is not the size of) unless you need to say what size of dog it is the same size as (since the size of dogs varies widely).