Let $\mathcal L$ be an <i>infinite</i> signature and $\mathcal A$, $\mathcal B$ two <i>finite</i> $\mathcal L$-structures such that
for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal A\models\varphi\iff\mathcal B\models\varphi.$$
> Does it follow that $\mathcal A$ and $\mathcal B$ are isomorphic?

Clearly, for finite signatures $\mathcal L$ the answer would be "yes".