Size is a hand-wavy way of saying it. What we're really asking is given two sets A and B, is there a function from A to B such that no two elements from A are mapped to a single element in B?

It turns out if you use ZFC (in particular choice) then one direction must hold. Either such a function exists from A to B or from B to A. This means that we can order all sets using the existence of these functions, and so in that sense size is a "universal property."