Let $\text{Top}_*$ be the category of pointed topological spaces, and fix some $(X,x)\in\text{Top}_*$. Suppose that for all $Y\in\text{Top}_*$ we can give a group structure to the set $[X,Y]$ of homotopy classes of maps fixing the base point, in such a way that there exists a functor $$ \mathcal F:\text{Top}_*\to\text{Gr},\; (Y,y)\mapsto[(X,x),(Y,y)] $$ where $\text{Gr}$ is the category of groups.
Is it true that the identity of $[(X,x),(Y,y)]$ is the class of the constant map?
If the functor $\mathcal F$ happens to be the most natural one, i.e., the one given by composition: for a continuous map $f:Y\to Z$,
$$\mathcal F(f):[X,Y]\to[X,Z],\quad[h]\mapsto[f\circ h],$$ then this is easily seen to be true. This is, for example, what happens for the fundamental group. However, what if $\mathcal F$ is an arbitrary functor (assuming, of course, that a functor not of this form exists, which for all I know it might not at all)? Is it possible that the identity of group is not the constant map?
Apologies, I have not fully read the question. This answer presumes that the functor lifts the normal Hom functor on both objects and morphisms.
Clearly the identity element of $[X,*]$ is constant. The image of the identity of a group under a group homomorphism is the identity of the codomain. Since, $*$ is initial and the composition of any constant map with another map is constant, we conclude that all identities are represented by constant.