What is the Relationship between a Closed Map and a Closed Linear Operator?

It's just a coincidence. They are both derived from "closed" in the sense of closed sets, but they are not directly related to each other.

As Eric Wofsey pointed out, the terms are just coincidence. A closed mapping and a closed linear operator are not equivalent when considering a linear operator between two Banach spaces.

To show that they are unrelated, you might want to do the following exercises, which is Exercise 1.74 in Megginson's An Introduction to Banach Space Theory:

• Let $X,Y$ be normed spaces and let $T : X \to Y$ be a linear operator that is neither injective nor the zero operator. Find a closed subset $F \subset X$ such that $T(F)$ is not closed in $Y$.
• Find a linear operator $T : X \to Y$ that satisfies the hypotheses of the closed graph theorem even though there is a closed subset $F \subset X$ such that $T(F)$ is not closed in the range of $T$.