Htykuut

Are inclusions in some way better than arbitrary injections? Are maps to sets of equivalence classes somehow better than arbitrary quotients?

This is a simple-minded answer, but I would say yes because there is a proper class of distinct injections into $Y$ --- the injecting set $X$ can live anywhere in the set-theoretic universe --- whereas inclusions into $Y$ are effectively just subsets of $Y$. Under the natural notion of equivalence for injections, distinct subsets give inequivalent injections, so you could say that going from injections to inclusions is a matter of selecting one distinguished, canonical element from each equivalence class of injections.

I would say that the injections into $Y$ are classified by the subsets of $Y$, and similarly the quotients of $X$ are classified by the equivalence relations on $X$.

It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".

Given a set $X$, in general there is a proper class of injections $Y to X$. However, many of these are isomorphic, where injections $i : Y to X$ and $j : Z to X$ are isomorphic if there is an isomorphism $k : Y to Z$ such that $i = j circ k$. The isomorphism classes of injections into $X$ are the subobjects of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a well-powered category). It is pesky to work with set-many proper equivalence classes, so we instead look for a set $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X to Y$ is in $P(Y)$ and $j : Y to Z$ is in $P(Y)$, we would expect $j circ i : X to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y to X$ and $j : Z to X$ are in $P(X)$, and there is (a unique) $k : X to Z$ such that $i = j circ k$, then $k$ is in $P(Z)$.

We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$. This is not the only choice of such representative inclusions, but it's a pretty good one.

We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).

The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a well-copowered category. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X to X/{sim}$, or just the set of all equivalence classes on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.

The nLab page you're looking for is called factorization systems. Here is my favorite one, which I think answers your question. In any category with finite limits and colimits, every morphism $f : X to Y$ has a canonical factorization

$$X to text{coim}(f) to text{im}(f) to Y$$

where $text{im}(f)$, the regular image, is the equalizer of the cokernel pair of $f$ (this is the "nonabelian" version of "kernel of the cokernel") and $text{coim}(f)$, the regular coimage, is the coequalizer of the kernel pair of $f$ (again, the "nonabelian" version of "cokernel of the kernel"). These two constructions are categorically dual and so, among other things, the coimage-image factorization of $f$ in the opposite category is the same sequence of maps but in the opposite order.

In $text{Set}$, the coimage and image are both the image of a function in the usual sense, but computed in different ways, which I think match the distinction you're getting at. $text{coim}(f)$ is computed, more or less, by constructing the equivalence relation on $X$ defined by $x_1 sim x_2 Leftrightarrow f(x_1) = f(x_2)$, then quotienting $X$ by it. $text{im}(f)$ is computed in a categorically dual way, although it looks a little strange at first: by first constructing the pushout $Y sqcup_X Y$, then isolating the subset of $Y$ of elements which are sent to the same element by both of the canonical maps $Y to Y sqcup_X Y$.

In particular, the factorization you want for an injection is the regular image factorization, and the factorization you want for a surjection is the regular coimage factorization, so they are in fact categorically dual. The full coimage-image factorization combines these.

It's a nontrivial theorem that the map $text{coim}(f) to text{im}(f)$ is an isomorphism in $text{Set}$. It's also an isomorphism in any abelian category and in $text{Grp}$ (this is an abstract form of the first isomorphism theorem), but in general it's just both a monomorphism and an epimorphism. A very instructive example is $text{Top}$, where $text{coim}(f)$ is the set-theoretic image topologized as a quotient of $X$, and $text{im}(f)$ is the set-theoretic image topologized as a subspace of $Y$. (Note that these coincide for compact Hausdorff spaces!)

Andrej's answer is a good one for explaining "what is special" about injections and quotients. In terms of formalizing the notion of "subset inclusion", David Roberts has already mentioned in a comment the notion of M-category, which could be used to represent "the category $mathrm{Set}$ together with the information about which injections are inclusions" (or which surjections are quotients).

Another formalization which is specifically adapted to "inclusion-like" subcategories is called (appropriately enough) a system of inclusions (with enhancements that add adjectives such as "directed" and "structural"); it was introduced in this paper by Awodey, Butz, Simpson, and Streicher. One could presumably dualize this somehow to obtain a notion of "system of quotients". Note that a system of inclusions is a particular kind of M-category.

As has been noted, when the category ${rm Set}$ is treated "purely category-theoretically" it does not "notice" which injections are inclusions, i.e. this information does not transfer usefully across an equivalence of categories. An M-category is a way of defining a "category-theoretic" structure (namely, a certain kind of enriched category) that nevertheless can carry this sort of information.

"Are inclusions in some way better than arbitrary injections?"

No they are not.

Anything that you can say about the one, you can say about the other.

In particular:
Given $X$, there is a set of inclusions into $X$. And there is a (large groupoid equivalent to a) set of injections into $X$. For all practical purposes these are the same thing. So they should be treated as the same thing.

Are quotient maps somehow better than arbitrary surjections?

Again no.

Same story.

    The fact that we can talk about inclusions versus injections is an artefact of our set theoretic foundations of mathematics. It would be better to use a language that disallows us from asking whether two sets are equal, and only allows us to talk about isomorphisms between sets.
    We largely already do this in practice. When we write $mathbb Z subset mathbb Q$, we don't stop to explain that elements of $mathbb Q$ are defined as pairs of elements of $mathbb Z$, and we don't distinguish between $mathbb Z$ and its image in $mathbb Q$. We just write $mathbb Z subset mathbb Q$.

    Unfortunately, I'm not knowledgeable enough in foundational issues in order to make sense of the notion of "a language that disallows asking whether two sets are equal, and only allows to talk about isomorphisms". But I'm sure that such a thing exists (and that it can be tweaked so as to be exactly as powerful as ZFC).