= Solution
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. Hence
$$
\boxed{M\#S\cong N\#S,\qquad M\not\cong N.}
$$
This is <failure of cancellation for connected sums of surfaces>.
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