[contravariant]

There is a reason mathematicians prefer to talk about 'cardinality' rather than size.

Anyway if you want a set to be at least as big as its subsets and consider them to be of equal size when they're isomorphic then you kind of end up with cardinality as a notion of size. In some sense it's simply the best notion of size we have if all you have is the structure of sets.

There are of course other structures you could choose, like topological spaces, vector spaces etc. Those can fail to be isomorphic even when the underlying sets are, so you get a richer notion of 'size'.