What is so special about categories that lead people to use them to "formalize math"? There are countless interesting structures - lists, trees, maps, graphs. Yet, categories - which, if I understand, is just a graph with some constraints on its shape - are apparently special somehow, in that people propose using it to "formalize all of math".
What is so different about this specific data structure that makes it so special?
 A: Category theory allows us to prove theorems that apply to many different mathematical theories. For example, the categorical theorem that says that left adjoint functors commute with colimits can be applied to prove that the tensor product is right exact and to prove that the suspension functor commutes with wedge products. Those results can be (easily) proved without category theory, but the spirit of modern math is to work with maximal generality in order to simplify the theories. 
Moreover, it is fundamental for comparisons of different theories. For example, it is well-known that a module over a group algebra is the same as a representation of this group, or that the homotopy theory of simplicial sets is the same as the homotopy theory of spaces, but these statements make the more sense when expressed in terms of category theory.
Finally, many modern theories wouldn't exist without category theory. For example, modern homotopy theory revolves around the notions of model, higher or enriched categories. Without category theory, these notions would be very difficult to define/use.
And there are a lot of other things that I didn't mention that show that category theory is, at the moment, the right framework to do "general mathematics".
A: It isn't merely the data structure that makes categories "special" but the operations that can be performed on them.  Functors, in particular, relate corresponding elements in different categories and can show how apparently different mathematical structures are—generally at a very abstract level—in fact the same.
