In relations $R^0$ is Always the Identity relation (Equality relation as you termed). We can prove it in 3 ways.
1. Just By-heart it and think of it as an Axiom of relations.
2. We know in Relation theory $R^n.R^0 = R^n$ where $.$ is Composition Operator (Composition/Composite of two relations)
We could even write $R.R^0 = R$, Now you need to think what should be $R^0$ in such a way that When any relation is Composed with it, results in the same relation. (Sounds like Identity element for Composition of relation)
3. Answer Why in General mathematics $n^0 = 1$ where $n \neq 0$. If you can answer that, You can answer why $R^0$ is always the Identity relation.
Try it.