The important thing here is that this is about a quasi particle, something that behaves sort of like a particle but is not (like a hole in the electron distribution), in a 2D crystal lattice. This only happens because the lattice is not isotropic, you see a different pattern depending in the direction you look, so having properties change with direction is not totally unexpected. We already have materials with anisotropic thermal conductivity for example.

This won’t happen in vacuum as vacuum is isotropic.

It depends which metric definition are you using. The one I wrote is a pseudo-Riemannian metric that is not positive defined.

Normally physicists use that generalized metric definition because spacetime in most cases has a metric signature of (-1, 1, 1, 1). Points with zero distance are not necessarily the same point, they just are in the same null geodesic.

You can make something like this properly by defining a different metric. For example with metric dl² = dx² - dy² the vector (1, 1) has length 0, so you can make a “triangle” with sides of lengths 1, -1 and 0.