In the Sketches of an Elephant, there is the following notion:
We define an object $C$ of a quasitopos $\mathcal{E}$ to be coarse if 'it cannot detect the lack of balance of $\mathcal{E}$; i.e. if, whenever $f: A \to B$ is both monic and epic, each morphism $A \to C$ factors uniquely through $f$.
It seems that the notion of coarse object can be given for an arbitrary category. Why, in general, such a notion does not appear anywhere, but only when $\mathcal{E}$ is a quasitopos. A possible answer is: because on quasitopos coarse objects have interesting properties not holding in general. Ok, it seems reasonable, but, if such a notion is not expressed elsewhere, how to study it?
Next, in general, I do not understand why Johnstone specifies that each morphism $g: A \to C$ factors uniquely: in fact, if $g=h \circ f=h' \circ f$, then uniqueness is ensured by the fact that $f$ is an epic. Is my argument correct?
The argument in the last paragraph is correct. I guess the motivation for writing the definition like this is that isomorphisms are characterized by this unique lifting property (for all morphisms). The context deals with mono- and epimorphisms which are not isomorphisms. So it is only natural to keep that unique lifting property as is. Not every definition has to be minimal. For example, did you know that in the definition of a ring it is superfluous to require commutativity of addition? It follows automatically. But no one cares, since it isn't useful to omit it.