> Why shouldn't that person get the higher wage? That person is also part of a society.
Ideally, they should. The problem comes in when a capitalist plays a labor arbitrage game that causes economic damage to real people in his home area, just so he can pocket the difference in labor costs.
Questions like...
>>> If your work can't be differentiated from someone elses, why should you command a higher wage?
...are pretty much always asked from the perspective of a labor-buying capitalist looking to maximize her own profits, regardless of externalities. Those questions have an insidious pseudo-moral tone that holds up the capitalist's profit over other concerns, and that subtly paints a worker's desire to make a respectable local wage as selfish.
It's very rare to turn the question around (at least from a position of power) to challenge the capitalist's decision-making.