Let $(\Omega, \mathcal F, \mathbb P)$ be probability space and $(X, \mathcal M)$ be measurable space. We define random measure $\mu$ as function $$\mu:\Omega \times M \to [0, \infty]$$ such that \begin{align} &\forall B \in \mathcal M &\mu(-, B)&:\Omega \to [0, \infty] \quad \text{is random variable,} \\ &\forall \omega \in \Omega & \mu(\omega, -)&: \mathcal M \to [0, \infty] \quad \text{is a measure}. \end{align} I heard about another approach to random measures (measure valued random variables) but I'm interested only in this case now.
The idea is very useful for Levy processes, because they can always be decomposed as sum of Wiener process, drift and limit of integral involving some random Poisson measure. But this definition still seems kinda odd to me - we are mixing 2 worlds together, and (due to lack of more natural or joint conditions I guess) require our function to behave nicely when one argument is fixed. There is no connection between axis whatsoever.
I was wandering, what random measure is in category theory? It feels like it could be a morphism from some sort of product on objects of different categories. Is it a common thing to join totally different objects in such manner? Markov kernels are definitely alike (almost the same), but outside of this example and stochastic processes?
I can offer two answers to your question. The first is an answer you have explicitly not asked for, but I think it does in fact address your comment about how "this definition still seems kinda odd to me - we are mixing 2 worlds together, and ... there is no connection between axis whatsoever". The second is an answer that is closer to what you have directly asked for, and is less complicated than just linking to Tobias Fritz's monograph, but (imo) is not as informative on its own.
(For the first point, I work in a slightly more restrictive setting than yours, because the explanation is less cumbersome.)
Firstly, let's see why the definition of random measure that you give does indeed come from the "measure valued random variable" definition. Given a Polish space equipped with its Borel sigma-algebra $(X,\mathcal{B})$, we consider the space $\mathcal{P}(X)$ of Borel probability measures on $X$, and equip this space with the narrow topology. This topology also provides us with a Borel $sigma$-algebra on $\mathcal{P}(X)$. One of several equivalent characterizations of the narrow topology is that it is the weakest topology on $\mathcal{P}(X)$ such that for every open set $U\subseteq X$, the map which takes a probability measure $\mu$ and returns $\mu(U)$ is continuous from $\mathcal{P}(X)$ to $[0,1]$. A corollary of this (using for instance the monotone class theorem) is that for every Borel set $B\subseteq X$, the map $\mu \mapsto \mu(B)$ is Borel measurable from $\mathcal{P}(X)$ (with the Borel $\sigma$-algebra coming from the narrow topology) to $[0,1]$.
Now, consider a measurable map $\kappa$ from $(\Omega, \mathcal{F})$ to $(\mathcal{P}(X),\mathcal{B})$. Then $\kappa_\# \mathbb{P}$ is a probability measure on $\mathcal{P}(X)$. Now, by composing the map $\kappa$ with the map $\mu \mapsto \mu(B)$, it automatically holds that $\kappa(\cdot)(B)$ is a measurable function from $\Omega$ to $[0,1]$, simply because the composition of measurable maps is again measurable.
In other words, the "odd" definition where we "mix 2 worlds together" actually comes directly from putting a reasonable measurable structure on the space of measures and considering measure-valued random variables. (I have only indicated why the measure-valued RV definition implies the "odd" definition, but the converse is also worth thinking about.) If you really want to think about exactly what I have just described but while using the word "category" more, you can say that the category we consider is $\mathbf{Pol}$ whose objects are Polish spaces with Borel sigma-algebras, and whose morphisms are Borel measurable maps. Then, the fact that for any Polish space $X$, the space $\mathcal{P}(X)$ (with the narrow topology) is again a Polish space, we can identify a random measure with a specific morphism from a Polish space $\Omega$ to another Polish space $\mathcal{P}(X)$.
As for a more directly category-theoretic perspective, you can do the following: define a category whose objects are measurable spaces and whose morphisms are stochastic kernels (a.k.a. random measures but the initial space $\Omega$ is not necessarily thought of as a probability sample space). One checks by hand that you can compose stochastic kernels and that this composition is associative, and that the map $x \mapsto \delta_x$ is the correct identity morphism. This category is known in the literature, see here: https://en.wikipedia.org/wiki/Markov_kernel#Composition_of_Markov_Kernels_and_the_Markov_Category.
I should mention that there is a connection between these two answers, but it probably uses too much category theory to be easily digested. But briefly, the idea is to look at the space $\mathcal{P}(X)$ as a "more intrinsically categorical" object (i.e. it does not just happen to also reside in the category of Polish spaces, alongside X, by accident), and then ask what is the intrinsic category-theoretic way to talk about $\textrm{Hom}(\Omega,\mathcal{P}(X))$ inside the category of Polish spaces (or more generally inside a suitable category of measurable spaces). The first relevant notion is that $\mathcal{P}(X)$ can be identified as a monad, namely the "Giry monad" (https://ncatlab.org/nlab/show/Giry+monad). With this perspective, there is a general category-theoretic notion for this situation where one studies morphisms where the codomain involves a monad. This notion is called a "Kleisli category", and it turns out that stochastic kernels are precisely the morphisms of the Kleisli category associated to the Giry monad, see here: https://ncatlab.org/nlab/show/monads+of+probability%2C+measures%2C+and+valuations#kleisli_morphisms_random_maps. (You can find all this in Tobias Fritz's monograph, which Kurt G. linked you to, and much more of course.)