Suppose we have two formal axiomatic theories of mathematics and the objects and operators in one theory can be defined in terms of the objects and operators in another theory such that the axioms of the former are theorems in the latter. Is there a term for this type of relationship? Where would I go to learn more about questions about this type of relationship? My intuition is that goal of something like ZFC is to be a theory that's as simple and general as possible such that any other mathematics we were interested in could be reframed in terms of ZFC, something one might call a universal theory although I don't know what the actual terminology is. Maybe there are widely different types of universal theories. Perhaps ZFC or even specific universes within ZFC can be constructed from ZFC itself.
1 Answer
You are searching for the concept known as interpretation.
 A model $N$ is interpretable in another model $M$ (perhaps a totally different language signature) if inside $M$ we can define a domain (perhaps a quotient by a definable equivalence relation on $k$tuples) and define the fundamental relations and structure of $N$ on that domain, in such a way that the defined structure is isomorphic to $N$.
For example, the complex number field $\langle \mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field, by representing $a+bi$ with pairs of real numbers $(a,b)$. However, it turns out that the real number field is not interpretable in the complex field.
A theory $T$ is interpreted in another theory $S$, if we have a uniform manner of defining a model of $T$ inside any model of $S$.
Two models (or theories) are mutually interpretable, if each is interpreted in the other.
A strictly stronger notion is biinterpretation. Two models are biinterpretable, if they are mutually interpretable, in such a way that the models can define the isomorphism of themselves with their copy inside (the copy of) the other structure. By iterating the interpretations, one obtains a fractal like situation:
My recent paper
 Alfredo Freire, Joel David Hamkins, "Biinterpretation in weak set theories," https://arxiv.org/abs/2001.05262.
contains an elementary introduction of interpretation of models of set theory, including the proof that no two extensions of ZF set theory are biinterpretable.
Here is a talk I gave for the Oxford set theory seminar on the topic: "Biinterpretation of weak set theories"

1$\begingroup$ Wow this exactly what I was looking for, thank you. $\endgroup$ Commented Jul 13, 2021 at 12:21

2$\begingroup$ I'm so glad about that. (By the way, it is possible to "accept" an answer, if you like, by clicking on the check mark to the left of the answer. This indicates one's general approval of the answer and let's everyone know that the question is in some sense settled.) $\endgroup$ Commented Jul 13, 2021 at 13:13

$\begingroup$ Typo, should be: lets. $\endgroup$ Commented Jul 13, 2021 at 15:02