Are the moon phases the same in the northern and southern hemispheres on the same day?

Strictly speaking, the moon phases you see should be different.

Suppose that two observation points on the earth are 10000 km apart (space linear distance, not the curve distance of the earth's surface) and observe the moon at the same time, the distance between the earth and the moon is about 380000 km, and the vertex angle of an isosceles triangle with two points as the base and the moon as the vertex is about 1.5 degrees, so that the observed moon phase between them is 1.5 degrees.

You can do an experiment, put a 3 cm ball 3 meters away from you, hit it with a beam of light, and close your left eye and right eye respectively. In the absence of other reference objects, it is basically impossible to distinguish the light and shadow changes on the ball through two eyes.