From definition of an equivalence relation it easily can be proved that an equality relation denoted "$=$" is indeed an equivalence relation, it means that every two arbitrary elements on a set $X$ with respect to $=$ are related to each other if they have the same value, therefore the set created by "$=$" on $X$ contains ordered pairs $(x,x)$ where $x \in X$.
On the other hand an identity relation over $X$ (which is a homogeneous binary relation ) denoted $\text{Id}_{X}$ is also contains ordered pairs $(x,x)$ where $x \in X$.
I think equality is a special case of an identity relation, for example we say "equality is the finest equivalence relation" I think it's true because an equality equivalence relation is actually an identity relation, and since identity relation is contained in all of equivalence relations hence equality is also contained in all of them, but can we say that :
is Every arbitrary identity relation also an equivalence relation?
It is easy to see that the "identity relation" and "equality" relation are the same. To say that the identity relation contains only those elements of type $\left( x, x \right)$, is the same as saying that $x$ is related to $y$, through the identity relation if and only if $x = y$.
Since you have already proven that equality relation is an equivalence relation, the identity relation also becomes an equivalence relation.