It is not necessarily true that Borel equivalence relations that are potentially in the same pointclass are Borel bireducible. For example, consider the orbit equivalence relations of the logic action of $S_{\infty}$ on the standard Borel space of torsion-free abelian groups of rank $n$. Then these orbit equivalence relations are essentially countable and hence are potentially $\mathbf{\Sigma^0_2}$ (for example, see this theorem). However, Simon Thomas proved that the Borel complexity of isomorphism of torsion-free abelian groups of rank $n$ (strictly) increases with rank (in this paper).

It is not necessarily true that Borel equivalence relations that are potentially in the same pointclass are Borel bireducible. For example, consider the orbit equivalence relations of the logic action of $S_{\infty}$ on the standard Borel space of torsion-free abelian groups of rank $n$. Then these orbit equivalence relations are essentially countable and hence are potentially $\mathbf{\Sigma^0_2}$ (for example, see this theorem). However, Simon Thomas proved that the Borel complexity of isomorphism of torsion-free abelian groups of rank $n$ (strictly) increases with rank (in this paper).