Zermelo-Frankel set theory for algebraists
We say that a mathematical theory is categorical if it has exactly one model, up to isomorphism.
We intend some theories to be categorical, for instance the Peano axioms for natural numbers, Euclid's planar geometry, and set theory. Other theories are designed not to be categorical, i.e., the theory of a group, the theory of a ring, etc.
You are asking whether there are general theorems about categoricity, and whether in particular the Zermelo-Fraenkel set theory is categorical. First we have:
Theorem: If a theory expressed in first-order logic is categorical, then it axiomatizes a unique (up to isomorphism) finite structure.
Thus Zermelo-Fraenkel set theory and Peano arithmetic are not categorical. In fact, they both have many models:
Theorem: (Löwenheim-Skolem theorem) If a theory expressed in first-order logic has an infinite model, then it has an infinite model of every cardinality.
How should one react to these results? Perhaps we need not worry about it. So what if there are many models of Peano arithmetic and set theory? If we can accept the fact that there are many different groups, why not accept the fact that there are many different set theories? The mathematical universe just gets richer this way (but the search for "absolute truth" has to shift focus).
We could also "blame" first-order logic for these undesirable phenomena. For instance, whereas Peano axioms do not "pin down" the natural numbers, the category-theoretic notion of the natural numbers object does: all natural number objects in a category are isomorphic (because they are all the initial algebras for the functor $X \mapsto 1 + X$). This is possible because the category-theoretic description speaks about the entire category, not just the object of natural numbers.
We can in fact do the same for set theory: assuming a suitable "category of classes", the Zermelo-Fraenkel universe of sets can be characterized (uniquely up to isomorphism) as a certain initial "ZF-algebra" (this point of view has been studied in algebraic set theory). Note however that from a foundational point of view we have not achieved much, as we just shifted the problem from sets to classes.
As an algebraist, should you be worried that the category of sets is not "uniquely determined" by the Zermelo-Fraenkel axioms? I don't think so. Algebra is generally quite robust, and works equally well in all models of set theory. Of course, there are also parts of algebra that depend on the set-theoretic ambient, but is that not a source of interesting mathematics?
The axioms of ZF are not axioms which are expressible in categorical terms - they involve not just objects and morphisms in $Sets$, but also the membership relation $\in$. Therefore it doesn't even make sense to ask if some category satisfies those axioms.
Instead, those axioms are satisfied by certain structures called models of ZF. A model is just some set (or class) $M$ together with a relation $\in_M$, which may or may not coincide with the usual membership relation, such that all of the axioms ZF1-ZF9 are satisfied when we restrict the quantifiers to range over elements of $M$ and we replace $\in$ with $\in_M$ in those formulas. Under the assumption of consistency of ZF, there are many models like that, of various cardinalities.
To relate back to categories, to any model $(M,\in_M)$ of ZF we can associate a category, let me denote it $Sets_M$, whose objects are elements of $M$ and whose morphisms are functions in $M$ (which, in the set-theoretic definition of a "function", are themselves some sets which in a suitable sense relate the domain and codomain). The category $Sets$ itself is what happens if you take $M$ to be $V$, the class of all sets, and $\in_M=\in$. Therefore, while the question as stated doesn't exactly make sense, hopefully this illustrates why the axioms of ZF do not determine $Sets$ uniquely in any sense.
You have to specify what you mean by a category 'fulfilling a list of axioms of set theory', because axioms of set theory are usually statements about sets in some logical language.
One reasonable interpretation would be to use the internal logic of a category to form an internal logical language, then formulate the axioms of set theory in the internal language and ask if they're true in the ambient category. Set can trivially internally formulate and satisfy all of the set-theoretic axioms of whatever background set-theoretic universe we're in, so in particular if we're working in $ZF$ then Set will satisfy ZF1-ZF9.
It turns out that other categories do this as well, so Set is not unique in this regard as other answers state. Specifically, Mike Shulman wrote a nice paper exploring how to extend the usual internal logic of a pretopos to a more general interpretation called the stack semantics, which allows us to formulate axioms involving unbounded quantifiers. Using the stack semantics of a topos, we can formulate all the axioms of $ZF$ internally and ask if they're true in the ambient topos.
Collection and replacement are satisfied in the stack semantics of any topos, but the separation axiom requires the introduction of a new topos-theoretic axiom schema -- any topos satisfying this additional axiom schema is called an autological topos, and these are precisely the categories which internally have the full strength of $ZF$ set theory.
So it seems like the answer to your question is:
No, Set is not the only category satisfying ZF1-ZF9 in the above sense, and the (bi)category of all categories satisfying ZF1-ZF9 would be precisely the (bi)category of autological toposes.
If you are looking for a unique characterization of Set, it is the terminal object in the category of Grothendieck toposes and geometric morphisms.
As mentioned in Andrej Bauer's answer it is also the initial 'ZF-algebra'. For a summary on this view with definitions you can take a look at these notes on Algebraic Set Theory by Steve Awodey, or take a look at the classical reference on Algebraic Set Theory by Joyal and Moerdijk for a full explanation.
Linked paper: arXiv:1004.3802 [math.CT]