Why and how can we use set theory with other undefined elements from logical theories? Consider the axioms of Hilbert geometry: the undefined elements are points, lines and planes. We are not talking about points as coordinates, lines as sets, etc. This theory has no model at all: we are not sure if we can describe points, ...,  as sets. However, in proofs we’ll be always using sets of points, sets of planes, etc, which $\mathbb{ZFC}$ does not explicitly describe as existent since its universe of discourse is limited to sets. Why can we create sets of points?
Since the original question was not totally clear, we can have an analogy with Peano’s naturals. If we use them, the naturals are not necessarily sets, but we can construct set of naturals. The same applies for the axiomatic construction of the reals. They are not necessarily sets and, from the universe of discourse of Zermelo’s theory, we can’t construct sets of reals unless reals are proven to be sets or we create new axioms that describe truths between real numbers (undefined elements) and sets.
The main focus of this question is the formality to describe the relation of the objects of different formal theories. How can we relate these objects, specifically sets and other structures? 
 A: It's analogous to how informal-but-precise algorithms can be performed by computer programs, or how (a la Godel's incompleteness theorem) statements about provability can be "expressed in arithmetic" in some sense even thought they're obviously not simply statements about natural numbers, or how computer games can "describe physics" even though the described objects aren't literally inside the computer: the point is implementation.
Basically, when we think of "doing geometry inside ZFC," we're not claiming (or at least, don't need to be claiming) that points, lines, etc. literally are "pure sets" - rather, we're claiming that there is a way to define within ZFC something that "behaves like" geometry in a precise sense. We're not even claiming that this implementation is unique, any more than one claimes that there is a unique computer program (or even a unique computer program in a given language) which performs a specific algorithm, or a unique method of Godel number, or similar.
I think a good example to work on understanding is the connection between natural numbers and finite ordinals. It's hard to argue that the natural number $17$ literally is the set it's represented by in ZFC (namely, the seventeenth finite ordinal), but on the other hand the finite ordinals together with the purely-set-theoretically-defined operations of ordinal addition and ordinal multiplication correspond to the "true" natural numbers in an obvious way (e.g. the ordinal sum of the first and second finite ordinals is the third finite ordinal, just like how $1+2=3$).
There is a lot to be said along these lines, but for now you should just think of the  following analogy: 

ZFC is to ordinary mathematics as your favorite programming language is to algorithms.

It's by no means the only way to implement mathematics in a unified way, nor are the "standard implementations" of various pieces of mathematics in ZFC (to the extent that they exist) the only such implementations, but that's not the point: we only need ZFC to do its job.
