Why is the slope of a horizontal line zero? Can you provide a geometric interpretation?

When two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0.

On a horizontal line, the y-value remains constant while the value of x changes indefinitely. Can we say the slope is 0 because, at the meeting place, the point is (0, y)?

Yes. Why I didn't see that is a sign of old age. LOL. I should know better than that as a college algebra self-study student.