Update: I'm marking this answer as community wiki so as not to "leech reputation."
OP here. I want to include some information I've gathered from answers and comments to this question (as well as several others) which I think most directly address what I was asking about.
I think the most succinct answer to my question is goblin's, where they write that
the axioms of group theory don't axiomatize the concept "element of a group." Rather, they axiomatize the concept "group." In a similar way, the axioms of ZFC don't axiomatize the concept "set." They axiomatize the concept "universe of sets" (or "von Neumann universe" or "cumulative hiearchy", if you prefer).
This is essentially what Hurkyl says in the following excerpt from their answer:
The point of things like the ZFC axioms isn't to define the notion of a set: it's to define the notion of a "universe (of sets)".
Each universe can be said to give a notion of "set", but that must be understood in the same sense that each vector space gives a notion of "vector".
Likewise, Carl Mummert writes in his answer to a related question, What is the definition of a set?,
A set on its own does not satisfy the ZFC axioms, any more than a vector on its own can satisfy the vector space axioms or a point on its own can satisfy the axioms of Euclidean geometry. [...] Instead of saying "a set is anything that satisfies the ZFC list of axioms", you need to start with the entire model of set theory. Then, it does make sense to say, for example, that a ZFC-set is an object in a model of ZFC set theory.
There are some circularity concerns that may arise while reading these answers. First is that of how we are able to discuss the manipulation of finite strings of symbols in first-order logic without having defined what a "set" is.
Addressing this issue, Stefan Geschke writes in his answer to a related question on MO:
The conclusion that I have arrived at is that you need to assume that we know how to deal with finite strings over a finite alphabet. This is enough to code the countably many variables we usually use in first order logic (and finitely or countably many constant, relation, and function symbols).
So basically you have to assume that you can write down things. You have to start somewhere, and this is, I guess, a starting point that most mathematicians would be happy with. Do you fear any contradictions showing up when manipulating finite strings over a finite alphabet?
Similarly, arjafi explains in their answer to When does the set enter set theory?,
Properly, logic shouldn't concern itself with notions of set. We can use the notion in an almost metaphorical sense, such as saying that our (first-order) language consists of a set of symbols, and from these symbols we define the set of formulae. This is not really a problem, since the basic objects of logic are symbols that can be written down, and our set or [sic] symbols can be just a listing of these symbols (or a description of how to write them), and our set of formulae is just a method for analysing a written expression and determining whether or not it is a formula.
However, a potential problem still arises when talking about models. Henning Makholm writes in the same answer I quoted when first asking this question that "if you want to do model theory on your first-order theory you need sets," and user21820 asks in a comment on Carl Mummert's answer I quoted above,
Just how do you define model? Aren't you working in set theory already with "set" as a given concept?
Some possible solutions can be found in the answers to a question I posed earlier on the site, Is the "domain of discourse" in axiomatic set theory also a "set"?. Rory Daulton suggests in his answer that
we have two concepts of "sets": those that can be described in the formal theory, and those that cannot but can be described in some "larger" theory. Yes, this is weird, but so is formal set theory.
Meanwhile, Peter Smith writes in his answer,
It is even better not to think of 'the domain' as a single object at all (whether 'set', 'class' or 'collection'). Instead of talking of a domain (singular) of individuals, just talk directly of the relevant individuals (plural). [...] If you want to formalize the metatheory in plural style, use a formal plural logic.
Finally, returning to my question of whether set theory can uniquely define a model of itself (which could then be considered to define the concept of a set in the same sense that a vector space defines the concept of a vector), user21820 explains in their answer,
The axioms of ZFC (or any other sufficiently strong first-order formal system) cannot define the notion of "set", in the sense that you're looking for, namely that ZFC cannot pin down a unique structure that satisfies ZFC. Why so? Because ZFC cannot prove its own consistency, by Godel's incompleteness theorem, and hence ZFC cannot prove that there is a model of ZFC. Furthermore, ZFC can prove that if ZFC is consistent then it has infinitely many models, not at all a single one.