Is the moon phase the same anywhere in the world?
Yes, everyone looking up at the moon from any spot on the globe sees the same moon, in more or less the same phase.
Can you give a technical reason why this is true?
The phases of our moon depends on the light emitted from the sun in relation with the shade cast by the earth upon the moon.
Since the moon orbits around earth and earth orbits around the sun we will see the same moon phase independently on where on the globe the observer is located.