Question: Do random variables form an example of a comma category?
Attempt: Let $(M_i, \mathscr{F}_i, \mu_i)$ denote an arbitrary measure space ($M_i$ is the point set, $\mathscr{F}_i$ is the $\sigma$-algebra, and $\mu_i$ is the measure), and let $(X, \mathscr{G})$ be a measurable space (i.e. no choice of measure).
If we let $\mathscr{M}$ denote the category of measure spaces and $\mathcal{M}$ denote the category of measurable spaces, then there is an obvious forgetful functor $\mathscr{M} \to \mathcal{M}$ which forgets the choice of measure.
If we consider $(X,\mathscr{G})$ with the identity to be a (one object, one morphism) category, then it is trivially a subcategory of $\mathcal{M}$ and thus we have an obvious inclusion functor $(X,\mathscr{G}) \to \mathcal{M}$.
Using the notation of the nLab page on comma categories, $C$ is the category of measure spaces $\mathscr{M}$, $D$ is the trivial category $(X, \mathscr{G})$, $E$ is the category of measurable spaces $\mathcal{M}$, $f$ is the forgetful functor $\mathscr{M} \to \mathcal{M}$, and $g$ is the inclusion functor $(X, \mathscr{G}) \to \mathcal{M}$.
Then the objects will be measurable functions $\psi:(M_i,\mathscr{F}_i, \mu_i) \to (X,\mathscr{G})$ or (perhaps equivalently) the measures induced on $(X,\mathscr{G})$ via $\psi$, i.e. $\mu_i \circ \psi^{-1}$, or the resulting measure space $(X, \mathscr{G}, \mu_i \circ \psi^{-1})$.
In particular, we can restrict $\mathscr{M}$ to be the subcategory of probability spaces, and take $(X,\mathscr{G})$ to be $\mathbb{R}$ with the Borel $\sigma$-algebra -- then the objects of the category are random variables, and (I think) two random variables are isomorphic in the category if and only if they have the same distribution.
The more general formulation in terms of measure and measurable spaces above might allow this to be extended to random measures, stochastic processes, and others.
This is almost a slice category, except that our agnosticism with respect to the measure on $(X, \mathscr{G})$ forces us to consider it as an element of the category of measurable spaces $\mathcal{M}$ rather than an element of the category of measure spaces $\mathscr{M}$, forcing us to resort to a more general comma category instead of a slice category.
I'm somewhat lost as to where you're trying to go with this, so I'll state my thoughts on the matter, hoping they're relevant. (although maybe this post says everything better)
First off, arrows are important, and if we follow our noses we find the category of measure spaces is a weird sort of thing — what we really want is a category whose arrows are the measurable functions, which means that the category $\mathcal{M}$ of measurable spaces is realy what we should be mainly interested in.
On $\mathcal{M}$, there is a functor $X \mapsto \operatorname{Meas(X)}$ (the name is not a standard one) that sends a measurable space $X$ to the set of measures you can define on it, and sends a map $\Phi:X \to Y$ to the pushforward function $\mu \mapsto \Phi_* \mu$.
The category $\mathscr{M}$ of measure spaces is, I think, the category of elements of $\operatorname{Meas}$, which reinforces my point of view that it should be understood as the functor $\operatorname{Meas}$ rather than as the category $\mathscr{M}$.
A random, real-valued variable on a measure space $X$ is nothing more than a measurable, real-valued function on the underlying measurable space; that is, $\mathcal{M}(X, \mathbb{R})$. The natural way to organize them across all spaces is via the the representable presheaf $\mathcal{M}(-, \mathbb{R})$.
That said, you can take the category of elements, which is equivalent to the slice category $\mathcal{M} / \mathbb{R}$, but again that is a sort of weird way to organize the data.
So yes, if you really like expressing things in terms of categories of elements, you can combine the projection $p : \mathscr{M} \to \mathcal{M}$ with the slice by $\mathbb{R}$ (which itself is the comma category $(\mathcal{M} \downarrow \mathbb{R})$) to get the comma category $(p \downarrow \mathbb{R})$, but it is a sort of weird way to express things.
Incidentally, IIRC, for any measurable space $(X, \mathscr{G})$, $\mathscr{G}$ is a locale, and you can even use ultrafilters to construct a topological space $(S_\mathscr{G}, \mathscr{G})$, so that measurable functions $(X, \mathscr{G}) \to \mathbb{R}$ are precisely the continuous functions $(S_\mathscr{G}, \mathscr{G}) \to \mathbb{R}$.
If you construct the corresponding topos of sheaves $\mathcal{E} = \operatorname{Sh(\mathscr{G})}$, then by the usual facts about sheaves on topological spaces, its real number object $\mathbb{R}_\mathcal{E}$ is the sheaf of real-valued continuous functions on $\mathscr{G}$.
In particular, if we equip $\mathscr{G}$ with a probability measure $\mu$, then the real numbers object $\mathbb{R}_\mathcal{E}$ is precisely the algebra of real-valued random variables on $(X, \mathscr{G}, \mu)$.