It's best to describe a space using perpendicular axis, though not required. Strictly speaking you could have a coordinate system which doesn't have perpendicular axis, but the maths gets more complicated.

Mathematically speaking this is asking what the optimum choice of basis is for a space, a 2-d manifold in the case of the earth. So it's mostly just convention, to make the mathematics easier.

It's been about a decade since I touched linear algebra, so I'm sure someone more recent can expand on my answer, but that's the basic jist.