Reference: Cancellability of the Dyck congruence
Your assertion is wrong, the congruence $\equiv$ is not cancellable from the right: for instance $\bar a a \bar a \equiv \bar a$, but $\bar a a \not\equiv 1$.

By the way the quotient $\Sigma^*/{\equiv}$ is not a group, but a monoid, called the polycyclic monoid, first introduced in [1].

[1] M. Nivat and J-F. Perrot, Une généralisation du monoïde bicyclique, Comptes Rendus de l'Academie des Sciences de Paris 271 (1970), 824-827.