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Take $M=T^2$, $N={\mathbb{RP}}^2\#{\mathbb{RP}}^2$ and $S={\mathbb{RP}}^2$. The first is orientable, while $N$, the Klein bottle, is not, so $M$ and $N$ are not homeomorphic.

Both $M\#S$ and $N\#S$ are nonorientable and have Euler characteristic $-1$. By the classification of compact surfaces, both are homeomorphic to the connected sum of three projective planes. HenceThis is failure of cancellation for connected sums of surfaces.