Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite 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".