Is there a bijection between mathematical structures and morphisms that preserve this structures forming a category? Is there a bijection between kinds of mathematical structures and morphisms that preserve all properties of this kind of structure?
For example:
Topologycal spaces $\leftrightarrow$  Homeomorphisms
Metric spaces $\leftrightarrow$  Isometries
Sets $\leftrightarrow$  Bijections
Groups $\leftrightarrow$  Isomorphisms of groups
Rings $\leftrightarrow$  Isomorphisms of rings
Modules $\leftrightarrow$  Isomorphisms of moodules

Edit:
Being more precise, my question becomes in 2 particular questions in concrete categories:


*

*Given a mathematical structure in a set (concrete object), ¿this follows which functions are the morphisms?.

*Given a function between sets, ¿this follows which objects must be the sets?.
 A: Your question is a little unclear for the reason Giovanni De Gaetano described.  Bracketing that, there are mathematical structures which are "preserved" by several types of isomorphisms.  The website nLab has this to say about Banach spaces, emphasis added:

In functional analysis, the usual notion of ‘isomorphism’ for Banach spaces is a bounded bijective linear map $f:V\to W$ such that the inverse function $f^{−1}:W\to V$ (which is necessarily linear) is also bounded. In this case one can accept all bounded linear maps between Banach spaces as morphisms. Analysts sometimes refer to this as the “isomorphic category”.
Another natural notion of isomorphism is a surjective linear isometry. In this case, we take a morphism to be a short linear map, or linear contraction: a linear map f such that $∥f∥\leq 1$. This category, which is what category theorists generally refer to as $\mathbf{Ban}$, is sometimes referred to as the “isometric category” by analysts.

So "isomorphism of Banach spaces" is not clear.  You might also find page 23 of The Joy of Cats (available for free here) interesting.  More generally, if you are interested in questions of this type, you should do some reading on category theory.
A: I don't know if it helps you, but one can define non trivial elementary embedding in ZFC, and the existence of such embedding is proved to be independant of ZFC : it's a possible axiom (see critical points for example). So I'm not sure if your question has a answer as you think it should...
