I think there’s a confusion here. When the outcome is multivariate, I think @mgrab is expecting to be able to write down a point-wise log likelihood such that each element of the multivariate outcome makes its own point-wise contribution. But it doesn’t work this way. The multivariate outcome isn’t factorizable in this way, because the likelihood associated with the ith element of the outcome is not conditionally (on the parameters) independent of the values taken by all the other elements of the outcome.
Note however, that in a model where you have a multivariate normal observation-level random effect, and then some other conditionally independent response distribution, now you can again talk about the point-wise likelihood per element. However, you will not likely be able to use PSIS-loo in this situation, because you will have a random effect parameter per observation, which will induce too much flexibility in the model for the PSIS approximation to the leave-one-out posterior to be reliable.
EDIT: this post is confusing at best and basically just wrong. See Model comparison between independent normals and multivariate normals