By the third isomorphism theorem, $S=\mathbb Z[x,y]/(x^2,y^2,2)\simeq \frac{\mathbb Z[x,y]/{(2x)}}{(x^2,y^2,2)/{(2x)}}$.
The numerator is isomorphic to $\mathbb Z_2[y]$ and the denominator is isomorphic to $(y^2)_2$.
It follows that $S\simeq \frac{\mathbb Z_2[y]}{(y^2)_2}$