Is the "modulo" in modulo operation and in congruence relations, the same?
I don't know how the definitions go but I'm going to try to define modulo operation in my own words.
Modulo operation: Let $n \in \mathbb{N}$ then a function $\cdot \bmod n : \mathbb{Z} \to \{0, 1, 2, \ldots, n-1\}$ given by $a \bmod n = r$ where $r$ is the remainder left when $a$ is divided by $n$, is called the modulo operation.
However, with that definition, I strongly suspect that "modulo" used in congruence relation is not the same. Because $a \equiv b \ (\bmod n) \Leftrightarrow n \mid (a-b) $ where $a,b \in \mathbb{Z}, n \in \mathbb{N}$ is a binary relation and not a function.
Am I correct in believing that they're not the same? I can certainly think how they can be connected but I don't think they're the same. For example: $a \equiv b \ (\bmod n) \Leftrightarrow a \bmod n = b \bmod n$
Second question: What does one mean when they say "modular arithmetic", are they referring to Modulo operation or the congruence relations?
Indeed, they are not the same. However, they are closely related. In fact, as you wrote,$$a\equiv b\pmod n\iff a\operatorname{mod}n=b\operatorname{mod}n,$$where the $\operatorname{mod}$ from the RHS is the module operation, whereas the one from the LHS is the one from congruence relations.
And modular arithmetic is about congruence relations.