More generally, I have proven that for quasi-compact and quasi-separated schemes $\mathrm{Qcoh}(X \times_S Y)$ is the bicategorical pushout of $\mathrm{Qcoh}(X)$ and $\mathrm{Qcoh}(Y)$ over $\mathrm{Qcoh}(S)$ in the bicategory of cocomplete linear tensor categories. The technique of the proof has many other applications as well.
Localizations of tensor categories and fiber products of schemes (arXiv:2002.00383)