For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an abelian category one can define a *categorically finitely generated* object as being an object M such that for any directed family of subobjects $M_\alpha \subset M$, such that the sum $\sum_\alpha M_\alpha$ is equal to the original $M$, then there exists an index $\alpha_0$ such that $M_{\alpha_0} = M$.

Now, in the category of $R$-modules this is easy to see, and the argument essentially relies on the fact that any module is the colimit of its finitely generated submodules. For quasi-compact and quasi-separated schemes the only reference I could find is Daniel Murfet's excellent notes. This is Corollary 64 from here.

I would like to know whether this result still holds for algebraic spaces or more generally for algebraic stacks. (I imagine that knowing this is essentially equivalent to proving Lemma 61 in Murfet's notes for a scheme but using the étale toplogy)

EDIT: Perhaps one cook up a variant of Proposition 15.4 of Laumon-Moret-Bailly. Unfortunately the argument there seems to use at the very end that a submodule of a finite type module is of finite type, which does not hold in the non noetherian case.

EDIT2: By the way, for qcqs algebraic spaces this was already known to Raynaud-Gruson [RG Proposition 5.7.8].