That's the metric for Euclidean three-dimensional space, and as such it's meaningful. Tensor methods are generally a bit of overkill for this trivial manifold, but it's educational to rewrite the metric in some truly bizarre coordinate system with off-diagonal elements as a demonstration of how the formalism works before taking on the non-trivial manifolds of GR.
However, it's not a "Newtonian metric" and it doesn't have much to do with Newtonian physics.
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