On the wikipedia page for converse relation it states:
If $I$ represents the identity relation, then a relation $R$ may have an inverse as follows:
A relation $R$ is called right-invertible if there exists a relation $X$ with ${\displaystyle R\circ X=I}$, and left-invertible if there exists a $Y$ with ${\displaystyle Y\circ R=I}$. Then $X$ and $Y$ are called the right and left inverse of $R$, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse $R^{–1}$ is used. Then $R^{–1}$ = $R^T$ holds.
My question is that doesn't the identity relation only make sense if it is a homogenous relation i.e: $R\subset S\times S$? If so then the above would suggest that all left and right inverses are equivalent, which doesn't seem right.
Yes, that's true.
But there is always such set $S$. Simply take $S=\operatorname{dom}(R)\cup\operatorname{rng}(R)$, this is sometimes referred to as the field of $R$ or $\operatorname{fld}(R)$.
If you have several relations, then you can simply take the union of their fields as your "set of interest".