Interesting observation. That's simply part of the usual definition of a magic square, and it does indeed feel arbitrary once you compare it with the hexagon case, which is very symmetrical. I first learned about magic squares from math books as a kid, about 25 years ago, and just accepted those rules as given.

Upon a quick research, there are many variations, and the closest to what you're describing is the "pandiagonal" magic square, where additional diagonals are considered (except they wrap around the edges of the square in a slightly funny way, so that every diagonal still contains exactly N numbers): https://en.wikipedia.org/wiki/Pandiagonal_magic_square