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.

Unique Existence and the Axiom of Choice
I'm currently working on a Django application using AngularJS for the frontend part.
I want for the moment to upload some images on the server and get as callback a list with the paths of the uploaded files. I want to send afterwards to the API (I am using Tastypie framework) as a POST request the callback and insert it into the database in a specific field. My issue is that I don't know how to approach the images upload (I should use PHP?) part and most important how to receive the callback? I hope that I explain clear enough. :D
