One assumption of the Poisson regression is that the target variable follows a Poisson distribution.
Do that also mean that each individual prediction given by a Poisson regression follows a Poisson distribution of parameter lambda = the predicted value y_hat ?
There are two main kinds of prediction -- a point prediction and a prediction interval.
Neither is a distribution but both relate to distributions.
If you're getting a point prediction, you will want some conditional statistic - typically a conditional expectation given the model.
If your prediction is an interval, you'll attempt to produce an interval that (under the assumptions) will achieve approximately the desired coverage. That will involve taking some account of the uncertainty/predictive variation resulting from the fact that the parameter values are estimated (as well as the variation in the distribution given the parameters).
Bayesian prediction -- here the prediction would typically be the expectation of the predictive distribution (but might be something else, like its mode), and an interval prediction would often be a HPD interval from the predictive distribution. (You could, produce the entire predictive posterior distribution as a prediction -- in simple cases with a particular form of gamma prior on the Poisson parameter I believe you obtain a negative binomial predictive distribution)
