In group theory, cosets of a normal subgroup are disjoint and all of the same size. For monoids, the analogy are congruence classes which partition the monoid. Are they also of equal size?
By the way, if you know some book where I could learn more about quotients of monoids with congruence classes, that would be great too.
No. For a simple example, let $M=\{0,1,2\}$ with the operation $\max$. Then there is a congruence relation with $\{0,1\}$ and $\{2\}$ as the congruence classes.