If I understand correctly, axioms for the real numbers do not determine real numbers completely, in the meaning that some other structures obey the same axioms.
But they do determine real numbers up to isomorphisms, meaning that those algebraic structures "resemble" real numbers, meaning that they are very similar constructs.
Since my background is mostly in physics and I do not know much about higher mathematical logic and theory of models, I would like to know how many these models there are?
In other words, how many structures isomorphic to the reals there are?
As far as "models up to isomorphism" the answer is simple. If $f\colon X\to Y$ is a bijection between two sets, then given any structure on $X$, we can define a structure on $Y$ such that $f$ is an isomorphism.
So the question is how many sets are of size $2^{\aleph_0}=|\Bbb R|$, and the answer is a lot of them. That is, given any non-empty set $X$, the class $\{Y\mid |X|=|Y|\}$ is a proper class. To see why, note that there is a proper class of sets, the by simply fixing some $x\in X$, we can define $X_a=(X\setminus\{x\})\cup\{a\}$. If $a\notin X\setminus\{x\}$ it is very easy to see that $|X_a|=|X|$, and if $X$ is infinite, as it is in the case of $\Bbb R$, then also when $a\in X$ and we have $|X_a|=|X|$. In either case a proper class.
Therefore there is a proper class of models which are isomorphic to $\Bbb R$. In fact, on each set of size $2^{\aleph_0}$ there are $2^{2^{\aleph_0}}$ different ways to define a structure isomorphic to the real numbers: every permutation will define a version of the real numbers, and that is just how many permutations there are on a set this size.
But we can also talk about the first-order theory of the real numbers, that is, the theory of real-closed fields, which is a decent theory that lets you even develop a smidgen of analysis (to some extent, but I cannot stress enough how small of a smidgen this is going to be). This theory is a first-order theory, and therefore does not include the axiom of completeness. Instead it merely states that every polynomial of odd degree has a root, which is also a way of stating the intermediate value theorem for polynomials.
As a first-order theory, it does not even determine the cardinality of the real numbers, and by the Löwenheim–Skolem theorems, it has models of every infinite cardinality. So that means that every infinite set can be made "a little bit like the real numbers" in some sense. But it's worse than this: while the second-order version of the reals, discussed above, had at least all the models on a set isomorphic to one another, here we can find a lot of non-isomorphic models on any infinite set. So the theory does not even describe the "isomorphism type up to cardinality".