Characterizations of infinite compact Abelian groups and probability spaces based on the forcing notion they give Let $G$ be an infinite compact Abelian group with the collection $\mathcal{B}$ of Borel subsets of $G$, and $m$ the (unique) normalized Haar measure on $\mathcal{B}$. This gives a natural forcing notion $\mathbb{P}_G$: for $A, B \in \mathcal{B}$ let $A \sim B \iff m(A \bigtriangleup B) = 0$, and let $\mathbb{P}_G$ consists of equivalence classes $[A]_\sim$ where $A \in \mathcal{B}$ and $m(A)>0$, with $[A]_\sim \leq [B]_\sim$  iff $m(A\setminus B)=0.$ 
Now define an equivalence relation $\equiv$ among infinite compact Abelian groups by
$G \equiv H \iff \mathbb{P}_G$ is forcing equivalent to $\mathbb{P}_H$.
Now my question is the following:

Question 1. What non-trivial facts one can say about $\equiv$ relation$?$ 

As a sample of explicit question, one may ask the following:

Question 2. Let $\kappa$ be an infinite cardinal and let $\mathfrak{g}_\kappa$ consists of compact abelian groups of size $\kappa.$ What is the cardinality of $\{[G]_\equiv : G \in  \mathfrak{g}_\kappa   \}?$

In general one can ask similar question for probability spaces. Let $(\Omega, \mathcal{F}, P)$ be an infinite probability space and assign to it, its natural forcing notion $\mathbb{P}_{(\Omega, \mathcal{F}, P)}$. Also define $\equiv_P$ relation between infinite probability spaces as above.

Question 3. What non-trivial facts one can say about $\equiv_P$ relation$?$

Similar to question 2, one may ask:

Question 4. Let $\kappa$ be an infinite cardinal and let $\mathfrak{p}_\kappa$ consists of probability spaces of size $\kappa.$ What is the cardinality of $\{[G]_\equiv : G \in  \mathfrak{p}_\kappa   \}?$

Remark. The equivalence of forcing notions is defined in the comments by Professor Hamkins.
 A: 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 $w(G)=w(H)$.
A: If you are looking for a sampling and analysis of definable forcing notions, I would definitely recommend taking a look at the book Forcing Idealized by J. Zapletal; the book is devoted to forcings of the form $Borel(X)/I$ for some (nicely definable) $\sigma$-ideal $I$.
Among other interesting things discussed are different measure algebras; like the ones you described.
Also I'll just add in passing that, it is consistent that the only weakly-distributive c.c.c. forcing notions are Maharam algebras (see for example, Velick2000.) 
