I actually assumed it would be easy to find a multivariate version of the Poisson distribution, but couldn't find any concrete solution (in terms of a well cited publication). It seems that multivariate Poisson models are not commonly used in business applications ("risk management", prediction of rare events). But why is this the case? Am I looking for the wrong thing? Any help is appreciated :)
Have a look at https://www.jstor.org/stable/2288540. The authors use a Dirichlet prior on the normalised intensities and a Multinomial likelihood on the counts. The posterior ends up being a Dirichlet distribution. For Bayesian analysts of the Multinomial Dirichlet distribution, have a look at https://people.eecs.berkeley.edu/~stephentu/writeups/dirichlet-conjugate-prior.pdf. Hope that helps.