From Category Theory by Mac Lane:
There is an equivalence between the categories: $\text{Set$_{f}$}$ and $\text{Finord}$ which is $(S, \#)$ where $S$ is the inclusion function from $\text{Finord}$ and $\#$ is the functor that maps a set to the finite ordinal having the same number of elements.
Equivalences allow us to compare categories which are "alike" but of very different sizes.
What example is the analogical comparison here?
Is the similarity: Set$_{f}$ has elements $\{X : \#X = n\}$ which all have the same cardinality so we can think of them as being "similar" to a single element in Finord ($n$)?
The analogy is that an equivalence of categories preserves many categorical properties. You can think of an equivalence of categories as being a homotopy equivalence of categories.
An equivalence of categories is the right notion of isomorphism of categories. It means that up to natural isomorphism the functors are inverses of each other. This means that equivalences of categories preserve limits and colimits.
Skeleta:
The example from the question is an example of the following general construction.
Consider a category $C$. A skeleton for $C$ is a (sub)category $S$ which is equivalent to $C$ such that every isomorphism class in $S$ contains a single object.
We can construct a skeleton for any category $C$ by taking a subcategory of $C$ containing only one object from each isomorphism class, $X_{[A]}$. Then for any $A\in [A]$ fix an isomorphism $\eta_A : A\to X_{[A]}$. Then we can define the functor $C\to S$ by sending $A$ to $X_{[A]}$ and $f:A\to B$ to $\eta_B\circ f \circ \eta_A^{-1}$.
It turns out this functor is an inverse equivalence to the inclusion $S\hookrightarrow C$.
Skeleta and equivalences:
It turns out that all skeleta for a category are not just equivalent, but isomorphic categories, and two categories are equivalent if and only if their skeleta are isomorphic. Thus you can think of an equivalence of categories as (sort of) saying that the categories differ in the number of copies of isomorphic objects.