Of course it's not necessary to make this identification, but it's fairly harmless since the groupoid of monomorphisms into an object $X$ is equivalent to the discrete category of subobjects, and it can be a slight technical convenience, especially in relation to smallness conditions. For example, we say that a category is well-powered if for each object the set of its subobjects is "small" (is a set), and this is convenient for example when discussing certain forms of adjoint functor theorems, etc.

In topos theory, assuming that a topos $E$ is well-powered (e.g., a Grothendieck topos), one way of describing a subobject classifier $\Omega$ is that the contravariant subobject functor $Sub: E^{op} \to Set$ is representable as $\hom(-, \Omega)$. This description is conceptually convenient to some people's taste.

Of course it's not necessary to make this identification, but it's fairly harmless since the groupoid of monomorphisms into an object $X$ is equivalent to the discrete category of subobjects, and it can be a slight technical convenience, especially in relation to smallness conditions. For example, we say that a category is well-powered if for each object the class of its subobjects is "small" (is a set), and this is convenient for example when discussing certain forms of adjoint functor theorems, etc.

In topos theory, assuming that a topos $E$ is well-powered (e.g., a Grothendieck topos), one way of describing a subobject classifier $\Omega$ is that the contravariant subobject functor $Sub: E^{op} \to Set$ is representable as $\hom(-, \Omega)$. This description is conceptually convenient to some people's taste.

Of course it's not necessary to make this identification, but it's fairly harmless since the category of monomorphisms into an object $X$ is equivalent to the discrete category of subobjects, and it can be a slight technical convenience, especially in relation to smallness conditions. For example, we say that a category is well-powered if for each object the set of its subobjects is "small" (is a set), and this is convenient for example when discussing certain forms of adjoint functor theorems, etc.

In topos theory, assuming that a topos $E$ is well-powered (e.g., a Grothendieck topos), one way of describing a subobject classifier $\Omega$ is that the contravariant subobject functor $Sub: E^{op} \to Set$ is representable as $\hom(-, \Omega)$. This description is conceptually convenient to some people's taste.

Of course it's not necessary to make this identification, but it's fairly harmless since the groupoid of monomorphisms into an object $X$ is equivalent to the discrete category of subobjects, and it can be a slight technical convenience, especially in relation to smallness conditions. For example, we say that a category is well-powered if for each object the set of its subobjects is "small" (is a set), and this is convenient for example when discussing certain forms of adjoint functor theorems, etc.

In topos theory, assuming that a topos $E$ is well-powered (e.g., a Grothendieck topos), one way of describing a subobject classifier $\Omega$ is that the contravariant subobject functor $Sub: E^{op} \to Set$ is representable as $\hom(-, \Omega)$. This description is conceptually convenient to some people's taste.