Let $\mathbb C[[t_1, ..., t_n]]$ be the local ring of formal power series in $n$ variables and let $\mathfrak{m}=(t_1, ..., t_n)$ be its unique maximal ideal. Let $M$ and $N$ be flat $\mathbb C[[t_1, ..., t_n]]$-modules. If there be an isomorphism of $\mathbb C[[t_1, ..., t_n]]$-modules (respectively of $\mathbb C$-modules) $$\mathbb C[[t_1, ..., t_n]]/\mathfrak{m}\otimes_{\mathbb C[[t_1, ..., t_n]]}M\cong\mathbb C[[t_1, ..., t_n]]/\mathfrak{m}\otimes_{\mathbb C[[t_1, ..., t_n]]}N,$$
can we conclude if there is an isomorphism of $\mathbb C[[t_1, ..., t_n]]$-modules $$M\cong N?$$
Consider the quotient field $Q$ of $\Bbb C[[t_1,\ldots,t_n]]$, then both $Q$ and $Q^2$ are flat modules such that the tensor product in question is zero, but $Q$ and $Q^2$ are not isomorphic as modules over $\Bbb C[[t_1,\ldots,t_n]]$.