You're making the same mistake as I sometimes do, thinking in terms of "object shape" or "functionality". Here, I use "subset" in a sense of "all values that belong to Sub also belong to Super", in line with a set-theoretic view of types in the post. Also see subsets on wiki: https://en.wikipedia.org/wiki/Subset

Try this one: A is subset of B iff all the values that belong to A also belong to B. In your example (and in real life), every dog is an animal, so dogs are a subset of animals.