Basically it comes down the treatment of non-connected covers / non-transitive actions.
Classical Galois theory – whether applied to field extensions or to covering spaces – focuses on connected covers and transitive group actions.
Every transitive left action of a (discrete) group $G$ is isomorphic to the set of left cosets of some subgroup of $G$ (with the obvious left $G$-action), which is how subgroups enter the picture.
One way to make this precise is to define connectedness abstractly.
Definition.
A connected object in a category $\mathcal{C}$ with finitary (resp. infinitary) coproducts is an object $X$ in $\mathcal{C}$ such that $\mathcal{C} (X, -) : \mathcal{C} \to \textbf{Set}$ preserves finitary (resp. infinitary) coproducts.
Remark.
According to this definition, an initial object is never connected.
Example.
A $G$-set is connected in the sense above if and only if it is transitive.
Example.
A ring $A$ is connected in $\textbf{CRing}^\textrm{op}$ considered as a category with finitary (!!!) products if and only if $\operatorname{Spec} A$ is a connected topological space.
(In this context, $\emptyset$ is not connected.)
Equivalently, $A$ is connected in $\textbf{CRing}^\textrm{op}$ if and only if $A$ has exactly two idempotent elements, namely $0$ and $1$.
Example.
A finite étale algebra over a field $k$ is connected in $\textbf{FÉt}_k{}^\textrm{op}$ if and only if it is a finite separable field extension of $k$.
So we can extract the objects studied in classical Galois theory, at least.
To get the actual posets is a bit more difficult.
This should not be surprising: after all, in the context of field extensions, this amounts to choosing an algebraic closure and embedding all the field extensions into that algebraic closure.
But it can be done: this is what the fibre functor is for.
Let $\mathcal{C}$ be a category and let $U : \mathcal{C} \to \textbf{Set}$ be a functor.
We may form the following category $\textbf{El} (U)$:
An object is a pair $(X, x)$ where $X$ is an object in $\mathcal{C}$ and $x \in U (X)$.
A morphism $(X, x) \to (Y, y)$ is a morphism $f : X \to Y$ in $\mathcal{C}$ such that $U (f) (x) = y$.
Composition and identities are inherited from $\mathcal{C}$.
Incidentally, $U : \mathcal{C} \to \textbf{Set}$ is representable if and only if $\textbf{El} (U)$ has an initial object.
In the case where $\mathcal{C}$ is the category of connected $G$-sets and $U$ is the forgetful functor, $\textbf{El} (U)$ is a preorder category, which can be canonically identified with the poset of open subgroups of $G$: just send $(X, x)$ to the stabiliser subgroup of $x$.
In the case where $\mathcal{C}$ is the opposite of the category of finite separable field extensions of $k$ and $U$ is the functor sending $K$ to the set of $k$-embeddings $\iota : K \to \bar{k}$, where $\bar{k}$ is a chosen algebraic closure of $k$, $\textbf{El} (U)$ is a preorder category, which can be canonically identified with the opposite of the poset of finite subextensions of $\bar{k}$: just send $(K, \iota)$ to the image of $\iota : K \to \bar{k}$.
Since Grothendieck's formulation asserts that the opposite of the category of finite étale $k$-algebras is equivalent to the category of finite $\textrm{Gal} (k)$-sets as categories equipped with fibre functors, restricting to the subcategory of connected objects and applying the construction above recovers the classical antitone isomorphism of posets.
