Forget GLMs for a second. Imagine you have measured some count variable for individuals across two groups, where the counts in each group are distributed as Poisson.

The first has population mean $\lambda_1$, and the second has population mean $\lambda_2$.

Is the variance the same for the two groups? No. The variance is constant when you keep $\lambda$ the same (stay in the same group) but if you move across to the other group, both the mean and variance change (naturally, since the variable is Poisson-distributed).

For a Poisson regression $\text{Var}(Y_i|\mathbf{x}_i) = \lambda_i$ (where $\lambda_i$ is a function of $\mathbf{x}_i\beta$), so the Poisson mean is different at different values of the predictors. Because the Poisson mean changes, of course the variance must change with it (since the variance is equal to the mean).