[This answer is an elaboration on what KConrad mentions in his comments below, posted while I was first writing this.]

At finite level one has a perfectly good theory of Frobenius elements attached to maximal ideals upstairs with ambiguity of inertia subgroups (so no ambiguity in the unramified case, but don't get too attached to that). This goes similarly at "infinite level" and one can in fact think about Frobenius elements in a "naive way" for $\overline{K}/K$ by working directly with the integral closure $O_{\overline{K}}$ of $O_K$ in $\overline{K}$.

That integral closure is quite gigantic (and not noetherian) but is really not as frightening as it may seem because it is the directed union of the rings of integers $O_{K'}$ of the subextensions $K'/K$ of finite degree, with those $O_{K'}$ quite concrete things. If you think about it, a prime ideal of $O_{\overline{K}}$ "is" nothing more or less than a compatible system of prime ideals $P_{K'} \subset O_{K'}$ (meaning $P_{K''} \cap O_{K'} = P_{K'}$ whenever $K' \subset K''$).

Let's see how this plays out with maximal ideals in $O_{\overline{K}}$ and $O_K$, and then with that experience under our belts we'll see that a "naive" approach to Frobenius elements works very well.
For a maximal ideal $M$ of $O_K$, the set $S_{K'}$ of maximal ideals in $O_{K'}$ over $M$ is a non-empty finite set, and we have the natural maps $S_{K''} \rightarrow S_{K'}$ defined by $\mathfrak{m} \mapsto \mathfrak{m} \cap O_{K'}$ (even surjective, but doesn't matter).
In this way $\{S_{K'}\}$ is an inverse system of non-empty finite sets, so the inverse limit $\lim S_{K'}$ is non-empty (no need for Zorn stuff here since these are all countable inverse systems).

But this inverse limit is precisely "the same" as a prime ideal $P$ of $O_{\overline{K}}$ satisfying $P \cap O_K = M$. For any such $P$, each prime ideal $P \cap O_{K'}$ is maximal (i.e., a nonzero prime ideal). Thus, $O_{\overline{K}}/P$ is a field, as it is the direct limit of the quotients $O_{K'}/(P \cap O_{K'})$ that are all fields, so $P$ is *maximal*. The upshot is that every maximal ideal $M$ in $O_K$ lifts to a maximal ideal $\overline{M}$ of $O_{\overline{K}}$.

Conversely, every maximal ideal of $O_{\overline{K}}$ is nonzero and hence its prime ideal intersection with *some* $O_{K'}$ is nonzero and thus maximal, whence its intersection with $O_K$ is maximal. So we conclude that $\overline{M} \mapsto \overline{M} \cap O_K$ is a surjection from the set of maximal ideals of $O_{\overline{K}}$ onto the set of maximal ideals of $O_K$. There is an evident action of ${\rm{Gal}}(\overline{K}/K)$ on the set of such $\overline{M}$ which permutes those over a common $M$. Let's next see that this Galois action is transitive.

If $\overline{M}$ and $\overline{N}$ are two maximal ideals of $O_{\overline{K}}$ over the same $M$ then the same goes for
the maximal ideals $\overline{M} \cap O_{K'}$ and $\overline{N} \cap O_{K'}$ for each finite $K'/K$, especially those which are *Galois*. But the usual story for number fields assures that the *finite* set $T_{K'}$ of elements of ${\rm{Gal}}(K'/K)$ carrying $\overline{M} \cap O_{K'}$ over to $\overline{N} \cap O_{K'}$ is *non-empty*, and for $K''/K'$ with $K''/K$ also Galois the natural map ${\rm{Gal}}(K''/K) \twoheadrightarrow {\rm{Gal}}(K'/K)$ clearly carries $T_{K''}$ into $T_{K'}$ (surjectively, but doesn't matter). So once again the inverse limit $$\lim T_{K'} \subset \lim {\rm{Gal}}(K'/K) = {\rm{Gal}}(\overline{K}/K)$$
is *non-empty*. But if you think about it, this inverse limit is precisely the set of $g \in {\rm{Gal}}(\overline{K}/K)$ carrying $\overline{M}$ over to $\overline{N}$. Hence, the fibers of the map $\overline{M} \mapsto \overline{M} \cap O_K$ of sets of maximal ideals consist precisely of ${\rm{Gal}}(\overline{K}/K)$-orbits on maximal ideals of $O_{\overline{K}}$. In that sense, all $\overline{M}$ over a given $M$ are "created equal" from the perspective of the Galois-action.

Now finally (!) we come to your question. For each maximal ideal $M$ of $O_K$ we choose a maximal ideal $\overline{M}$ of $O_{\overline{K}}$ over it, and define the maximal ideal $\overline{M}_{K'} := \overline{M} \cap O_{K'}$ of $O_{K'}$ over $M$ for each $K'/K$. Consider the stabilizer subgroup ("decomposition group")
$$D(\overline{M}|M) = \{g \in {\rm{Gal}}(\overline{K}/K)\,|\,g(\overline{M}) = \overline{M}\}.$$
The above transitivity shows that each of the natural surjections
${\rm{Gal}}(\overline{K}/K) \twoheadrightarrow {\rm{Gal}}(K'/K)$
with finite *Galois* $K'/K$ carries $D(\overline{M}|M)$ *onto* the decomposition group $D(\overline{M}_{K'}|M)$. But for each $K''/K'$ that is Galois over $K$ the natural map $D(\overline{M}_{K''}|M) \rightarrow D(\overline{M}_{K'}|M)$ is surjective by the usual theory, so in particular for the non-empty finite subset $\phi_{K'} \subset D(\overline{M}_{K'}|M)$ of "Frobenius elements" we get a natural surjection $\phi_{K''} \rightarrow \phi_{K'}$ induced by the restriction map ${\rm{Gal}}(K''/K) \twoheadrightarrow {\rm{Gal}}(K'/K)$.

Finally, consider the inverse limit $\lim \phi_{K'} \subset {\rm{Gal}}(\overline{K}/K)$. This is *non-empty* and consists of precisely those elements $g \in {\rm{Gal}}(\overline{K}/K)$ such that $g(\overline{M}) = \overline{M}$ and the induced map on the field $O_{\overline{K}}/\overline{M}$ has restriction to each finite subfield $O_{K'}/\overline{M}_{K'}$ is the $q$-power map for $q = \#(O_K/M)$. But that says $g$ induces the $q$-power map on $O_{\overline{K}}/\overline{M}$, so such $g$ certainly deserve to be called "Frobenius elements" (at $\overline{M}$). Note that given one such $g$, all others are related to it through multiplication against the subgroup $I(\overline{M}|M) = \lim I(\overline{M}_{K'}|M)$ ("inertia subgroup") of elements of $D(\overline{M}|M)$ that induces the identity on the residue field $O_{\overline{K}}/\overline{M}$.

So that finally is the definition of a Frobenius element in ${\rm{Gal}}(\overline{K}/K)$ relative to $\overline{M}$ over $M$: it preserves $\overline{M}$ and induces the map $t \mapsto t^{\#(O_K/M)}$ on the residue field $O_{\overline{K}}/\overline{M}$. The above shows that such elements exist, and for any $\overline{N}$ over $M$ conjugation against any choice (which we saw does exist!) of element of ${\rm{Gal}}(\overline{K}/K)$ carrying $\overline{M}$ to $\overline{N}$ clearly carries $D(\overline{M}|M)$ over to $D(\overline{N}|M)$ and carries Frobenius elements over to Frobenius elements. (With a tiny bit of work one sees that $O_{\overline{K}}/\overline{M}$ is algebraically closed, so it is even an algebraic closure of $O_K/M$, but this was
not needed above; indeed, $\overline{K}$ everywhere above could have been
an arbitrary Galois algebraic extension of $K$.) What could be better than that?