One of these ways (from which Cauchy-Schwarz and the other Hilbert Space results follow) is that d_2 is the only metric that satisfies the Parallelogram Law [1]:

2 d_2(x) + 2 d_2(y) = 2 d_2(x + y) + 2 d_2(x - y)

[1] https://en.wikipedia.org/wiki/Parallelogram_law

Remove the factors of 2 on the right, and square all the d_2.