Second part:
Since $\sim$ is an equivalence relation, we have $g\sim g$ and $g^{-1}\sim g^{-1}$. Further, as $\sim$ is congruence, and $h\in H \implies h\sim e \implies gh\sim ge=g$ and so $ghg^{-1} \sim gg^{-1}=e$ for any $g\in G, h\in H$
Here are two examples :
$1 - $ Consider the relation $\equiv$ ( an equivalent relation), then
$$a \sim b \Leftrightarrow a\equiv b \mod 2 $$
That is, $a$ and $b$ will be in the same class $\overline{a}$ if their remainders of the division by $2$ are the same. For example $4$ and $6$ belong to the same class, which we are going to choose a representant $0$, because
$$6 = 3 \cdot 2 + \color{red}{0} \ \ \text{and} \ \ 4 = 2 \cdot 2 + \color{red}{0}$$
then we say $\overline{4} = \overline{6} = \overline{0}$. If we think, there are two distinct classes: $$\overline{0} = \{x \in \mathbb Z ; x \equiv 0 \mod 2, \text{$x$ is even}\}\ \ \text{and}\ \ \overline{1} = \{x \in \mathbb Z ; x \equiv 1 \mod 2, \text{$x$ is odd}\}$$
The set of all classes is
$$\mathbb Z_2 = \{\overline{0}, \overline{1}\}$$
$2-$ Consider the relation
$$(a,b) \sim (c,d) \Leftrightarrow ac = bd $$
This equivalent relation gives us the fractions, that is the filed of fractions of $\mathbb Z$. Similarly we choose a class representant for example,
$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6 } = \cdots$$
we choose $\frac{1}{2}$ to be the class representant. Notice that $\mathbb Q = \{ \frac{a}{b} ; a,b \in \mathbb Z, \ \ \text{where}\ \ b \neq 0\}$ is the set of all classes.
Best Answer
"The discrete congruence" on a structure $X$ (and also "the discrete equivalence relation" on a set $X$) refers to the equality relation $=$ on $X$: $$\{(a,a)\mid a\in X\}.$$ This is the minimal congruence / equivalence relation on $X$. Its equivalence classes are singletons.
In contrast, the maximal congruence / equivalence relation on $X$ is called "the trivial congruence" (or "the trivial equivalence relation"): $$\{(a,b)\mid a,b\in X\}.$$ It has a single equivalence class, $X$.
I believe the terminology comes from an analogy with the maximal and minimal topologies on a set: the discrete topology (in which singletons are open) and the trivial topology (in which the only non-empty open set is $X$). The terminology, while not universal, is at least somewhat common, as a google search reveals:
https://www.google.com/search?q=%22the+discrete+equivalence+relation%22
https://www.google.com/search?q=%22the+discrete+congruence%22