Why does the metric tensor always relate to cartesian coordinates?
Let's take the simple case for the metric tensor in 3D-space without a time dimension,
$g_{ij}= \begin{bmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2\; \sin^2(\theta) \end{bmatrix}$
here, the $\sin^2(\theta)$ stems from the fact that we originally derived the distances in cartesian coordinates as $\rm ds=\sqrt{dx^2+dy^2+dz^2}$ and then know the transformation between cartesian and polar. So the exact form of $g_{ij}$ as function of it's target coordinates, is always derived from the original coordinates, which are the cartesian ones.
But why don't we describe the metric tensor based on some other original coordinates, like hyperbolic and transform then to spherical ones (apart from the fact that it would be ugly business)?
So cartesian coordinates seem in some way special, my first idea was that maybe because they're an inertial frame of reference they would provide a natural basis for GR. But this can't be the case, as differential geometry comes from pure math, which doesn't care about inertial/noninertial statements.
So what is going on, is it the fact that we simply 'discovered' math in euclidean space first and later learned how to relate different coordinate systems to the euclidean one?
Same question naturally extends to relativity and minkowski coordinates.