There is an article with a related question, giving a negative answer in general:
In https://arxiv.org/pdf/2007.06095.pdf there is a counterexample (due to G. Halmos) mentioned. In that counterexample three $\sigma$-algebras $\mathcal{A}, \mathcal{F}$ and $\mathcal{G}$ are constructed (quite involved), such that $\mathcal{A}\otimes(\mathcal{F}\cap\mathcal{G})\neq (\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})$ and in further consequence it is shown that the latter $\sigma$-algebra is never of product form.