What is a "geometric rectangle"?
Geometric rectangle is not a standard term. That's why they define it in the question.

In communication complexity, Alice has an input from the set $X$, and Bob has an input from the set $Y$. Usually we assume that the sets $X,Y$ are unstructured. A combinatorial rectangle is a subset of $X \times Y$ of the form $A \times B$, where $A \subseteq X$ and $B \subseteq Y$.

In this question, the sets $X,Y$ are ordered, and the authors defined a geometric rectangle to be a subset of $X \times Y$ of the form $A \times B$ where $A$ is a subinterval of $X$ and $B$ is a subinterval of $Y$. That is, $A = \{ x \in X : x_1 \leq x \leq x_2 \}$ for some $x_1,x_2 \in X$, and $B = \{ y \in Y : y_1 \leq y \leq y_2 \}$ for some $y_1,y_2 \in Y$.