A point is from two points if its distance from each point is the same?

It's the middle point of line.Hardly! How far would that be from a point 3/4 way along the line? Try centre of a circle, or of a sphere.

The freezing point

A circle in 2-dimensions, a sphere in 3-d.

This is the center of the circle. From the center of the circle, all the points on the circle are equally distant.

The set of all points in space that are the same distance from a given point is called a sphere.