Distances with non-euclidean metric

Hello,
when measuring length of geodesic shortest paths, or more in general, when measuring the length of a parametric curve in the space, what we usually do is to sum the length of infinitesimal arcs of that curve, assuming an euclidean norm.

Why this choice?
I have not found in literature any mention on the possibility of using other norms, like L1-norm.

Why not to allow to measure the length of infinitesimal arcs of a curve in $$\mathbb{R}^2$$ by doing instead:

$$ds = \left| \frac{\partial \mathbf{p}}{\partial x} \right| dx + \left| \frac{\partial \mathbf{p}}{\partial y} \right| dy$$

Thanks...