Define a distribution with the same mean and variance as having property S.
Obtaining a two-valued (discrete) distribution with property S is easy. Let $P(x=m+d)=P(x=m-d)=\frac12$ with $m>0$. The mean is obviously $m$ and the variance is $d^2$, so $d=\sqrt m$. Similar computations lead to more-valued discrete distributions with property S.
For continuous distributions, there is obviously the normal $\mathcal N(m,m)$ distribution, which indeed is sometimes used to approximate the Poisson distribution. The uniform distribution can be made to have property S; fixing the lower bound at 0 we must have $b/2=b^2/12$ for the upper bound $b$, leading to $b=6$.
The most general method of obtaining distributions with property S would be through the moment generating function, as suggested in comments.