@WlodekKuperberg (perhaps) and I (for sure) were exposed to this kind of examples by Karol Borsuk, or possibly Karol Borsuk simply had an example like the one I will present below:
\begin{equation} D\ :=\ \{z\in\mathbb C: |z|\le 1\}\ \subseteq\ \mathbb C \ \end{equation} \begin{equation} A\,\ :=\,\ D\times\{0\}\ \cup\ \{1\ \ \ i\ \ -\!1\ \ -\!i\}\times [-1;0]\,\ \subseteq\,\ \mathbb C\times\mathbb R \end{equation} \begin{equation} X\,\ :=\,\ A\,\ \cup\,\ \{-1\ \ 1\}\times [0;1]\,\ \subseteq\,\ \mathbb C\times\mathbb R \end{equation} \begin{equation} Y\,\ :=\,\ A\,\ \cup\,\ \{i\ \ \ 1\}\times [0;1]\,\ \subseteq\,\ \mathbb C\times\mathbb R \end{equation}
Then $\ X\ $ and $\ Y\ $ are homeomorphic while $\ X\times I\ $ and $\ Y\times I\ $ are not.
REMARK One may check Karol Borsuk's series of publications about the uniqueness of topological decomposition into Cartesian products, and a paper by Hanna Patkowska about the uniqueness of the decomposition of ANRs into 1-dimensional ANR.