Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give

If $G$ is a compact group with infinite weight, then the Maharam type of the Haar measure on $G$ is equal to its weight - See Theorem 2.4 in S. Grekas, On products of topological measure spaces, Handbook of measure theory, Vol. 1, edited by E. Pap, Elsevier 2002.

As the measure algebra of a compact group is Maharam homogeneous, Maharam's theorem implies that it is isomorphic to the usual measure algebra on $2^\kappa$, where $\kappa$ is the Maraham type. This is the usual complete Boolean algebra for adding $\kappa$ random reals.

Therefore: $G \equiv H$ (in your notation) iff $w(G)=w(H)$.