There is another extremely important way in which they are like the normal distribution: both are maximum entropy distributions, i.e. each is the "most generic" within their respective families of distributions.
So is Gamma, Binomial, Bernoulli, negative-Binomial, exponential and many many more. Maxent distribution types are very common. In fact the entire family of distributions in the exponential family are Maxent distributions.
[1] https://en.wikipedia.org/wiki/Poisson_distribution#Maximum_e...
[2] https://en.wikipedia.org/wiki/Normal_distribution#Maximum_en...