The definition of a category requires that there exists a morphism for each object $X$, $id_X$ such that for all morphisms $f$, $f\circ id_X=id_X\circ f = f$.
But it doesn't say that this morphism is unique.
Is an object in a category allowed to have multiple, distinct morphisms $id_1,id_2,...$ that all satisfy that for all morphisms $f$, $f\circ id_i=id_i\circ f = f$?
You have that $${\rm Id}_1 \stackrel{(1)}{=} {\rm Id}_1 \circ {\rm Id}_2 \stackrel{(2)}{=} {\rm Id}_2,$$where in $(1)$ we use that ${\rm Id}_2$ is a "right-identity" and on (2) that ${\rm Id}_1$ is a "left-identity". Just like for groups.