If you are satisified with your example of the cardinals as unique objects defined using choice then there is an easy answer to your question. Note that there is not a unique ordinal which is in bijective correspondence with each set; there are many, but there is always a least one which we call a cardinal. So the uniqueness comes from the well ordering of the ordinals. Given the axiom of choice you can always well order the domain of objects in which you are interested and then choose the least one. This will of course, be unique, but I doubt this is what you had in mind. But I think it does show that a better example than the cardinals is needed for uniqueness.
One can construct saturated models by transfinite induction and then show that, under certain circumstances, these are unique. One also has $\beta \mathbb{N}\setminus \mathbb{N}$ which needs choice to be non-empty, and it is also unique --- but probably also not what you had in mind.