Since the subspace topology is a special case of an initial topology, and you are asking about generalizations anyway, I'm going to describe how the initial topology can be expressed categorically.

Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor and let $(X_i)_I$ be a family of spaces and also, by abuse of notation, the corresponding functor from the indexing set $I$ to $\mathbf{Top}$. If we want to equip a set $S$ with the initial topology with respect to some family of functions $(f_i:S\to X_i)_I$, then what we actually want is a cone $(f_i:(S,\tau_S)\to (X_i,\tau_i))$ over $(X_i)_I$ in $\mathbf{Top}$ such that
$\text{Id}:(Uf_i:U(S,\tau_S)\to U(X_i,\tau_i))_I\longrightarrow(f_i:S\to X_i)_I$ is a universal arrow from $U$ to the cone $(f_i:S\to X_i)_I$.

Here, universal means that for each cone of spaces $(g_i:(Y,\tau_Y)\to(X_i,\tau_i))_I$ and each arrow from
$(Ug_i:U(Y,\tau_Y)\to U(X_i,\tau_i))_I$ to $(f_i:S\to X_i)_I$, which is the same as a set map $h:Y\to S$ such that $f_i\circ h=g_i$, there is exactly one continuous map $h':(Y,\tau_Y)\to(S,\tau_S)$ satisfying $f_i\circ h'=g_i$ (formally it is a map between cones) such that $\text{Id}\circ Uh'=h$. Now, this simply means that $h'=h$ and that the set map $h$ is actually a continuous map.

Replacing $\mathbf{Top}$ and $\mathbf{Set}$ by arbitrary categories gives the notion of a *strictly initial lift*. We can generalize further by allowing the $\text{Id}$ in the above description to be an isomorphism (*initial lift*) or even just a morphism (*semi initial lift*).

Have a look at this nLab article which describes the dual notion of a final lift.

There is also section V.9 *Adjoints in Topology* in Mac Lane's *Categories for the Working Mathematician*, which deals with this construction and how it can be used to construct the limits in $\mathbf{Top}$.