In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.
The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.
In modular arithmetic this property is having the same rest when divided by a prescribed integer
If $a=b\bmod m$ or $a\equiv_m b$ you essentially say that $a$ and $b$ are in the same equivalence class with respect to the equivalence relation $\equiv_m$.
So you could say that "$a$ and $b$ are in the same equivalence class when we look at the remainder upon dividing by $m$", which definitely longer and more cumbersome to say "$a$ congruent to $b$ modulo $m$" but nonetheless another way to express the same concept.