Suppose we have two categories $C,D$ with same objects. For every object $X$ in $C$ there is an identity arrow $1^C_X\in Hom_C(X,X)$. As $C$ and $D$ have the same object, for that same object $X$ there is an identity arrow $1^D_X\in Hom_D(X,X)$. Does this mean that $1^C_X=1^D_X$?
My guess is that this is not the case, because in this proof of identity arrows being unique, it is assumed we are talking about a certain category, so the identities of a category should not be able to interact with the other(if no additional structure is imposed), even if they have the same objects.
Is this correct? Do two categories with same objects share the identity arrow?
$\mathrm{Hom}_{\mathscr C}(X, X)$ and $\mathrm{Hom}_{\mathscr D}(X, X)$ are not even necessarily the same sets (and may not have a nonempty intersection). The hom sets are unrestricted by the set of objects. So the identity arrows on $X$ in the two categories is not required to be the same.
As you suggest, there is precisely one identity morphism on $X$ in $\mathscr C$, and precisely one identity morphism on $X$ in $\mathscr D$, but these identity morphisms are data pertaining to $\mathscr C$ and $\mathscr D$ respectively and, a priori, do not relate to one another.