Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language consisting of one or more non logical symbols (corresponding to $\in$)?
How to think about those systems, do they define the objects or the non logical symbols or both? Or are such systems merely statutes, rules for working with the objects and symbols? Could a system of this kind ever be categorical?
Edit: as the question is formulated Peano axioms for natural numbers and Tarski axioms for geometry are examples. Among others! But I thought I was asking for something else. But I think I learned something anyway, both from formulating the question and from reading the comments.
Perhaps you're looking for something like map theory, which is an attempt to provide a general foundation for mathematics where the fundamental objects are functions rather than sets?