This is a surprisingly subtle question!
Right off the bat, we have an obvious observation: if $\mathcal{M}$ is a "potential $\omega$" (= is isomorphic to the $\omega$ of some model of $\mathsf{ZFC}$), then $\mathcal{M}$ must satisfy all the arithmetic consequences of $\mathsf{ZFC}$. This set of sentences, call it "$\mathsf{ZFC_{arith}}$," is not very well understood but we can prove some basic things about it; see e.g. this old answer of mine.
That said, it's also worth noting that under a mild assumption we can whip up a formula that $\mathsf{ZFC}$ proves defines a model of $\mathsf{PA}$ which has no additional restrictions - see here - but this is quite different from the usual way arithmetic is implemented in $\mathsf{ZFC}$, which you are asking about.
But is that enough? That is, if $\mathcal{M}\models\mathsf{ZFC_{arith}}$, is $\mathcal{M}$ a potential $\omega$?
To see the issue, consider the following very artificial example. Let's say we replace the interesting arithmetic structure of the naturals with just the single equivalence relation "has the same parity as." Basically, this turns the naturals into two infinite blobs. Now consider a structure $\mathcal{A}$ in this language consisting of two infinite blobs of different cardinalities. Since the theory of two infinite blobs is complete, $\mathcal{A}$ satisfies everything $\mathsf{ZFC}$ proves about "the naturals as a pair of blobs." However, $\mathcal{A}$ is not isomorphic to any model of $\mathsf{ZFC}$'s "blobby naturals." This is because the $\mathsf{ZFC}$-theorem "the even and odd naturals have the same cardinality" does have consequences for the isomorphism type of "potential blobby naturals" but is not captured at the first-order level.
In fact, $\mathsf{ZFC_{arith}}$ falls well short of "potential $\omega$-ness" in a completely unfixable way. Barwise/Schlipf proved the following:
For a countable nonstandard $\mathcal{M}\models\mathsf{PA}$, the following are equivalent:
(Strictly speaking, they proved that any structure appearing in some non-$\omega$-model of $\mathsf{ZF}$ is recursively saturated. This gives the top-to-bottom direction, with the bottom-to-top direction being folklore if I understand the history correctly. Note that countability is actually only needed in the bottom-to-top direction!)
The omitting types theorem then prevents any first-order theory from capturing potential $\omega$-ness. (Thanks to Andreas Blass for pointing this out to me in the comments below!) Depending on whether you like recursive saturation, this may or may not answer your question in the specific case of countable models. Unfortunately (or interestingly), uncountable models pose much more difficulties, and I believe no characterization is currently known even of the size-$\aleph_1$ models of $\mathsf{PA}$ which are isomorphic to some $\mathsf{ZFC}$-models' version of the naturals. (But that shouldn't be too surprising, since even the ordertypes of size-$\aleph_1$ models of $\mathsf{PA}$ aren't classified yet.)
As a coda, I vaguely recall an old paper (by H. Friedman?) on the following problem: given a first-order theory $T$, when does a tuple of formulas $\Phi$ have the property that, if we let $T_\Phi$ be the set of sentences $T$ proves holds of the structure interpreted by $\Phi$, then for all $\mathcal{A}\models T_\Phi$ there is some $\mathcal{B}\models T$ with $\mathcal{A}\cong \Phi^\mathcal{B}$? Basically, these $\Phi$s should not be "missing" any structural data (e.g. the "blobby naturals" example above was missing the structural data of the $n\mapsto n+1$ bijection). I can't seem to find the paper at the moment, however. My recent mathoverflow questions 1, 2 address this general theme, current absence of a reference notwithstanding.
Best Answer
As it happens, this very point was the subject of a discussion John Baez and I had here. (Part of a long discussion on the $\omega$'s of models of ZF.) John's viewpoint is that "standardness" is a relative concept.
This question cannot be totally disentangled from philosophical issues (but there are some purely mathematical aspects, which I'll address below). If you're a strong Platonist (like Gödel), you believe there is a unique mathematical universe of sets "out there". Call it $V$; then by definition the standard $\omega$ is the $\omega$ of this $V$.
If you prefer to hedge your bets, but still take a more-or-less realist stand, then you might like what we called the "three-decker sandwich". There is some universe $V$, the "outermost" universe; standardness is relative to this. We make no ontological claims about $V$; maybe there are a variety of "universes" that could serve for $V$. Inside $V$ is a model $U$ of ZF, and $\omega^U$ is standard if it's isomorphic to $\omega^V$. The locution "the standard $\omega$" is rejected as unnecessary, and burdened with metaphysical assumptions.
Of course, if everything is developed relative to $V$, then it fades into the background. It's needed mainly to give meaning to the term "standard". (Also in locutions like "the standard $\in$".) The "three-decker sandwich" is $\omega^U\subset U\subseteq V$.
The three-decker sandwich helps explain how non-standard $\omega$'s "work". Consider $\omega^U$. We have $\omega^V\subset\omega^U$, indeed it's a proper initial segment. At the bottom of p.44, Halmos defines $\omega$ as the intersection of all the "successor sets" (aka inductive sets) contained in a set $A$. Now, $\omega^V$ is a successor set, but it does not belong to $U$. That's why $\omega^V$ is smaller than $\omega^U$, even though $V$ is larger than $U$. Informally, the "inhabitants" of $U$ cannot "see" $\omega^V$, and so they compute the intersection incorrectly.
What about the idea that $\omega$ consists of the elements that can be obtained by starting with $\varnothing$ and applying the successor function $n$ times, with $n\in$ "the standard model $\mathbb{N}$" of the Peano Axioms with 2nd-order induction? Halmos does not have a formal theorem expressing this. Rather, he says that $\omega$ provides a rigorous counterpart of the intuitive description that the natural numbers are 0,1,2,3, "and so on". Then in the next chapter, he shows that $\omega$ satisfies the 2nd-order Peano axioms. In a set-theory context, $\mathbb{N}$ is essentially defined to be $\omega$. The two notations emphasize different aspects of the set of natural numbers: $\omega$ as an ordinal, $\mathbb{N}$ as a free-standing structure, whose signature (in the sense of mathematical logic) would include + and $\cdot$ but not $\in$.
Back to philosophy. If you're some breed of formalist, then the universe of all sets is a fiction. All that matters is what you can prove in some formal system like ZF. In ZF, one can give a formula $\varphi(x)$ and prove that there is a unique set $w$ satisfying $\varphi(w)$; moreover, $\varphi(x)$ is a formalization of the usual definition for $\omega$. Statements mentioning "the standard $\omega$" are rephrased using $\varphi$.
No doubt there are many other philosophical positions one might take. Perhaps one accepts the "multiverse of universes" (like Joel David Hamkins), but feels all the “legitimate” ones have the same $\omega$. (This seems to be the view of Scott Aaronson.) The argument for this: our intuitions about the natural numbers seem much more concrete and solid than about the entire universe of all sets, which disappears into the misty heights with the higher cardinals. (And is fuzzy around the edges, thanks to things like the Continuum Problem.) Others might feel that there's plenty we don't know about $\mathbb{N}$; maybe the notion of "the standard natural numbers" is a bit foggy after all.
As for the textbooks, these usually keep philosophy to a mininum. A couple of quotes from Halmos' preface:
I don't know what Halmos' personal philosophy was, but these quotes suggest a stylistic Platonism. As long as we're working in some fixed "outermost" set-theory universe $V$, we're entitled to use the notion of "the standard $\omega$". Pedagogically this is fine: it seems to have a clear intuitive meaning, and if we're not discussing non-standard models, might as well let sleeping dogs lie.
An analogy. Quantum mechanics boasts a large body of controversy over its proper interpretation. David Mermin once noted that most physicists in practice go by the "shut up and calculate" interpretation. With regard to these set-theory matters, I dare say most mathematicians are happy with "shut up and prove theorems" most of the time.