There won't be any property which really distinguishes $\mathcal{O}_X$ inside $\mathsf{Mod}(X)$, since any invertible $\mathcal{O}_X$-module $\mathcal{L}$ induces an auto-equivalence of categories $\mathcal{L} \otimes -$. Instead, we may hope for properties of invertible $\mathcal{O}_X$-modules (which only have to be proven for the special case $\mathcal{O}_X$ as soon as they are categorical). I hope that you don't mind that I switch to $\mathsf{Qcoh}(X)$ when appropriate, because $\mathsf{Mod}(X)$ is too large and does not really incorporate the condition that $X$ is a scheme.

1. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then every epimorphism $\mathcal{L} \to \mathcal{L}$ is an isomorphism.

2. The endomorphism ring $\mathrm{End}(\mathcal{L})$ is commutative. In particular, the group $\mathrm{Aut}(\mathcal{L})$ is commutative.

3. If $X$ is quasi-compact and quasi-separated, then $\mathcal{L}$ is a finitely presentable object of $\mathsf{Qcoh}(X)$.

4. If $X$ is separated, then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a direct sum of copies of $\mathcal{L}$. I don't know a classical reference for this (anyone?), but it is proven in Proposition 3.18 here (let $I=0$ there).

5. We know that there is an open covering $\{X_i \to X\}$ such that $\mathcal{L}|_{X_i}$ is a projective generator of $\mathsf{Qcoh}(X_i)$ (also satisfying 2)). This can be formulated categorically as follows (see here) when $X$ is quasi-separated: There is a family of thick subcategories $\mathcal{T}_i \subseteq \mathsf{Qcoh}(X)$ with $\bigcap_i \mathcal{T}_i=0$ such that the images of $\mathcal{L}$ in the quotient abelian categories $\mathsf{Qcoh}(X)/\mathcal{T}_i$ are projective generators.

There won't be any property which really distinguishes $\mathcal{O}_X$ inside $\mathsf{Mod}(X)$, since any invertible $\mathcal{O}_X$-module $\mathcal{L}$ induces an auto-equivalence of categories $\mathcal{L} \otimes -$. Instead, we may hope for properties of invertible $\mathcal{O}_X$-modules (which only have to be proven for the special case $\mathcal{O}_X$ as soon as they are categorical). I hope that you don't mind that I switch to $\mathsf{Qcoh}(X)$ when appropriate, because $\mathsf{Mod}(X)$ is too large and does not really incorporate the condition that $X$ is a scheme.

1. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then every epimorphism $\mathcal{L} \to \mathcal{L}$ is an isomorphism.

2. The endomorphism ring $\mathrm{End}(\mathcal{L})$ is commutative. In particular, the group $\mathrm{Aut}(\mathcal{L})$ is commutative.

3. If $X$ is quasi-compact and quasi-separated, then $\mathcal{L}$ is a finitely presentable object of $\mathsf{Qcoh}(X)$.

4. If $X$ is separated, then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a direct sum of copies of $\mathcal{L}$. I don't know a classical reference for this (anyone?), but it is proven in Proposition 3.18 here (let $I=0$ there).

5. We know that there is an open covering $\{X_i \to X\}$ such that $\mathcal{L}|_{X_i}$ is a projective generator of $\mathsf{Qcoh}(X_i)$ with a commutative endomorphism ring (and this characterizes invertible modules $\mathcal{L}$). This can be formulated categorically as follows (see here) when $X$ is quasi-separated: There is a family of thick subcategories $\mathcal{T}_i \subseteq \mathsf{Qcoh}(X)$ with $\bigcap_i \mathcal{T}_i=0$ such that the images of $\mathcal{L}$ in the quotient abelian categories $\mathsf{Qcoh}(X)/\mathcal{T}_i$ are projective generators with a commutative endomorphism ring.

There won't be any property which really distinguishes $\mathcal{O}_X$ inside $\mathsf{Mod}(X)$, since any invertible $\mathcal{O}_X$-module $\mathcal{L}$ induces an auto-equivalence of categories $\mathcal{L} \otimes -$. Instead, we may hope for properties of invertible $\mathcal{O}_X$-modules (which only have to be proven for the special case $\mathcal{O}_X$ as soon as they are categorical). I hope that you don't mind that I switch to $\mathsf{Qcoh}(X)$ when appropriate, because $\mathsf{Mod}(X)$ is too large and does not really incorporate the condition that $X$ is a scheme.

1. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then every epimorphism $\mathcal{L} \to \mathcal{L}$ is an isomorphism.

2. The endomorphism ring $\mathrm{End}(\mathcal{L})$ is commutative. In particular, the group $\mathrm{Aut}(\mathcal{L})$ is commutative.

3. If $X$ is quasi-compact and quasi-separated, then $\mathcal{L}$ is a finitely presentable object of $\mathsf{Qcoh}(X)$.

4. If $X$ is separated, then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a direct sum of copies of $\mathcal{L}$. I don't know a classical reference for this (anyone?), but it is proven in Proposition 3.18 here (let $I=0$ there).

There won't be any property which really distinguishes $\mathcal{O}_X$ inside $\mathsf{Mod}(X)$, since any invertible $\mathcal{O}_X$-module $\mathcal{L}$ induces an auto-equivalence of categories $\mathcal{L} \otimes -$. Instead, we may hope for properties of invertible $\mathcal{O}_X$-modules (which only have to be proven for the special case $\mathcal{O}_X$ as soon as they are categorical). I hope that you don't mind that I switch to $\mathsf{Qcoh}(X)$ when appropriate, because $\mathsf{Mod}(X)$ is too large and does not really incorporate the condition that $X$ is a scheme.

1. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Then every epimorphism $\mathcal{L} \to \mathcal{L}$ is an isomorphism.

2. The endomorphism ring $\mathrm{End}(\mathcal{L})$ is commutative. In particular, the group $\mathrm{Aut}(\mathcal{L})$ is commutative.

3. If $X$ is quasi-compact and quasi-separated, then $\mathcal{L}$ is a finitely presentable object of $\mathsf{Qcoh}(X)$.

4. If $X$ is separated, then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a direct sum of copies of $\mathcal{L}$. I don't know a classical reference for this (anyone?), but it is proven in Proposition 3.18 here (let $I=0$ there).

5. We know that there is an open covering $\{X_i \to X\}$ such that $\mathcal{L}|_{X_i}$ is a projective generator of $\mathsf{Qcoh}(X_i)$ (also satisfying 2)). This can be formulated categorically as follows (see here) when $X$ is quasi-separated: There is a family of thick subcategories $\mathcal{T}_i \subseteq \mathsf{Qcoh}(X)$ with $\bigcap_i \mathcal{T}_i=0$ such that the images of $\mathcal{L}$ in the quotient abelian categories $\mathsf{Qcoh}(X)/\mathcal{T}_i$ are projective generators.