Two-point perspective is just a special case of three-point perspective in which the camera is oriented parallel to the horizontal plane. SketchUp's two-point perspective mode is a fast and exact way to point the camera that way (you could do it manually if your orbit skills are good enough). That's why you lose the mode when you orbit any direction except perfectly horizontally. As noted, all this orientation does is to cause vertical (parallel to blue axis) lines not to converge in the view. Parallel to any other plane will cause the same effect for lines perpendicular to that plane.
When viewing real 3D objects, our two-eyed visual system blends depth and direction of view with perspective and we interpret converging edges as really being parallel. Tricking that system is the basis of many optical illusions.
But on a 2D image or computer screen there is no binocular depth available and view direction is toward the screen no matter what the model orientation in the view, so our visual system can't do the interpretation correctly. As a result, three-point perspective can look exaggerated and unrealistic, especially when viewing an object from up close or with a wide angle camera. Two-point perspective tends to give the visual system an easier task to interpret the underlying 3D.