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I have found a proof of this result 3 years ago and almost the finished writing everything up (heavy example of procrastination). The result is that for $S$-schemes $X,Y$, where $X,Y,S$ are quasi-compact and quasi-separated, indeed $\mathrm{Qcoh}(X \times_S Y)$ is the bicategorical pushout of $\mathrm{Qcoh}(X)$ and $\mathrm{Qcoh}(Y)$ over $\mathrm{Qcoh}(S)$ in the bicategory of cocomplete tensor categories. Actually, the technique of the proof has many other applications as well. I will link the paper here as soon as I have finished it.