Preamble: My knowledge on mathematical logic is very limited but I think this question deals with some concepts in logic.
It seems the Peano axioms and axioms for real numbers using Dedekind cut uniquely characterize the natural and real numbers. The question is that does Hilbert axioms for Euclidean geometry do the same for the Euclidean geometry? If yes how.
What is the mathematical importance of having such an axiomatic system (an axiomatic system that uniquely characterizes the model)?
Is there a term used in mathematical logic for axiomatic systems that uniquely characterize the model? I need to know the terminology to help me find the right references on the topic.