Given a scheme $X$ and divisor $D\subseteq X$, you can take the line bundle $$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \mathcal{M}$$ where $\mathcal{M}$ is the sheaf of rational functions on $X$. It satisfies $\mathcal{O}_X(D+D') =\mathcal{O}_X(D)\otimes\mathcal{O}_X(D')$. Moreover, under weak conditions these are the only coherent sheaves with an inverse under $\otimes$.

One categorical level up, you can consider sheaves of categories on $X$, with $\text{QCoh}_X$ being the analogue of $\mathcal{O}_X$.

My question is: what is the analogue "$\text{QCoh}_X(D)$" of $\mathcal{O}_X(D)$?

If at all possible, the definition should mimic the above, defining it as a subsheaf of categories of $j_*\text{QCoh}_\eta$ defined by similar [conditions], and with the same identity for $D+D'$. Ideally, under the same weak conditions as above, these are the only invertible sheaves of categories.