Here is a general definition of the word "coupling" that covers every use I've seen of it. (And this generality is necessary because sometimes one does not define a coupling on an exact product space, but some subset of it,)
Let $\Lambda$ be an index set of arbitrary cardinality, and $f_\lambda: (\Omega, \Sigma) \rightarrow (\Omega_\lambda, \Sigma_\lambda)$ be a family of measurable maps, and $\mu_\lambda$ be a family of probability measures defined on $(\Omega_\lambda, \Sigma_\lambda)$. We say that $\mu$ is a coupling of the $\mu_\lambda$ via the maps $f_\lambda$ if $\forall \lambda \in \Lambda$ we have $f^*_\lambda(\mu)=\mu_\lambda$. The usual situation is that $\Omega$ is some measurable subset of the product of the $\Omega_\lambda$ and the maps are the coordinate projections. (With the required hypotheses on $\Lambda$ and the $\Omega_\lambda$ in order to be able to use Kolmogorov extension when it's necessary.)
This recasting of coupling makes it appear pretty category theoretic. Is there any category theoretic significance here, or any results in probability theory that are derived from category theoretic ramblings about coupling?
Recastings of various notions from probability theory, such as the one you are interested in, or of probability theory as a whole, are regularly put forward, in various degrees of detail, by category theorists. It is probably (!) fair to say that these approaches are considered with a moderate to non-existent degree of interest by most probabilists.
For a discussion of the reasons of this lack of excitement, for some quite elaborate examples of such recastings by category theorists, and for all the qualifications the sweeping statement in the preceding sentence of this post requires, one can peruse the MO question Is there an introduction to probability theory from a structuralist/categorical perspective? and its answers.