I know that the concept of being isomorphic depends on the category we are working in. So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or geometry, I often hear that people say that such a structure is complete in the sense that any other set that satisfy their properties is isomorphic. What is the real meaning and implications of being isomorphic? At least to me it is not very clear.
Let's see for example the axioms of geometry. To begin with, there are different systems of axioms, all of them trying to define that object called geometry. So if we are talking about the same object, I suppose the theories generated should be isomorphic each other. Does this make sense? Again what is the real implication of being isomorphic? According to the definition there should be a bijection between the sets, which means that it cannot be larger sets (with a higher cardinality, or even different cardinality) that hold the propierties of the given structure and also the operations defined on each structure should be mantained in the bijection. Is this all? Please, if I'm saying bullshit forgive me I just want to really understand this concept. I will really appreciate any comment.
Edit: In my example I meant Euclidian Geometry.