What shall we say... it will be correct when the distribution of the response is Poisson and when the samples are paired (when pairs of samples share a common source of variance).

The Poisson distribution has a strict relation between mean and variance and should be used only when this relationship is given or makes sense (that is, the data were generated by a Poisson process and not by something else). Often, real data shows strong indications of over-dispersion, which means that the variance is considerable larger than the mean. In such cases you should consider the negative-binomial (Pascal) distribution or a quasi-Poisson model (which includes a free variance parameter at the cost of not providing a strict likelihood anymore, which is probably not of interest to you anyway).

Hello Horacio Salomón Ballina-Gómez

I think it's good to have a look at these Assumptions.

You can see more in this link

https://statistics.laerd.com/spss-tutorials/poisson-regression-using-spss-statistics.php

Abolfazl Ghoodjani , Horacio Salomón Ballina-Gómez clearly said that the samples are paired. Therefore, your 3rd point already does not fit. In case of dependent samples you might consider generalized linear mixed models, to account for the dependencies via the random intercept term.

Rainer Duesing Yes, thank you, you are absolutely right.

Horacio,

you want to compare two paired samples that are distributed according to a Poisson distribution. or you want to compare two paired samples with Poisson GLM.