Given two measurable spaces $(X,\mathscr{F}),(Y,\mathscr{G}),$ $f:X \to Y$ measurable and $\mu:\mathscr{F} \to [0,\infty)$ a measure, the pushforward of $f_*(\mu):\mathscr{G} \to [0, \infty)$ is defined as $f_*(\mu)(G)=\mu f^{-1}(G).$
Is this terminology coincidental, or is this an example of a universal object in some category? If so, which category, and why is it universal?
Let me elaborate on the comment by @tomasz. Consider a category $\mathtt{Mes}$ where objects are measurable spaces (sets endowed with $\sigma$-algebras) and morhpisms are measurable maps. For each object $M$ you can construct another object $\mathcal M(M)$ whose elements are measures on $M$, and the $\sigma$-algebra is generated by the evaluation functions $\theta_A:\mathcal M(M)\to \Bbb R$ given by $\theta_A:\mu\mapsto \mu(A)$ for each measurable subset $A$ of $M$. You can regard $\mathcal M$ as an endofunctor on $\mathtt{Mes}$, and its action on morphisms is exactly given by the pushforward construction for measures.
Regarding your original question: I guess that the notion of pushforward for measures was called that way due to the natural idea that you push the measure from one space to another along some map. I have only basic skills in category theory, and I am not much familiar with the pushout and pullback constructions in its general setting, however from what little I know it seems that pushouts are often applied to formalize the idea of gluing sets together. In that case, even if there exists a category for which pushforward of measure is described as a category-theoretical pushout, I am afraid it may be a rather artificial construction without much insight. In a sense, the naming may indeed be just a coincidence, which does not yet rule out the possibility of a connection, just maybe this connection maybe an artificial and purely technical one, existing due to another coincidence.