That distribution is still a point estimate for a multinomial, not truly the distribution of your certainty in that estimate itself. This is essentially a generalization of logistic regression, which will of course give the probability of a binary outcome, but in order to understand the variance of your prediction itself you need to take into account the uncertainty around your parameters themselves.

This can be done for neural networks, through either bootsrap resampling of the training data or more formal bayesian neural networks, both of these are fairly computationally intensive and not typically done in practice.