This question has been asked by Teimuraz Pirashvili many years ago. I forgot about it after a while and remembered only now by accident. He probably knows the answer by now, but I still don't.

In the category of modules over a ring $R$, the module $R$ is a projective generator. This property does not determine it uniquely, even up to isomorphism, but it does when $R$ is commutative: it can be reconstructed as the ring of endotransformations of the identity functor. (For noncommutative $R$, it is still true that the category is equivalent to the category of modules over the endomorphism ring of any of its projective generators that may exist.)

Now what can be said about the structure sheaf $\mathcal O_X$ of a scheme $X$? Can it be detected inside the category of $\mathcal O_X$-modules without using any additional structure? Of course it is the unit of the monoidal structure, but can it be also characterized as an object in a plain category, without invoking any additional structures? Can in fact the category of $\mathcal O_X$-modules possess other non-isomorphic monoidal structures? Can it be equivalent to the category of $\mathcal O_Y$-modules for some other scheme $Y$? Or, say, some analytic space, or whatever?

Note that $\mathcal O_X$ is not even a generator - the subcategory generated by it is the category of quasicoherent sheaves. (**NB** As @HeinrichD points out in the first comment, even that is wrong - this is only true locally, in the appropriate sense.) In any case, also in this category $\mathcal O_X$ is typically not projective. So the same question arises - is it some particular kind of generator?

There are of course several reconstruction theorems but I still cannot figure out which (if any) of them provide answers to these questions.

LATER - as pointed out by Will Sawin, I should be more clear. Rather than modifying the question (I don't quite understand how) let me try to formulate it once more:

Consider the object $\mathcal O_X$ of the abelian category of sheaves on a scheme $X$. What are its properties formulable without the use of tensor product or any other additional structures, just as an object of this abelian category?

For example, as Will Sawin explains in his answer, a commutative ring inside its category of modules cannot be distinguished among rank one projectives. But at least it is a projective generator with a commutative endomorphism ring (in fact it is isomorphic to that ring), which is a very restrictive property. Are there some similar properties of $\mathcal O_X$?