Why does stress-energy tensor come from metric variation and can we derive other conserved quantities in a similar way?

The usual classical mechanics version of the stress-energy tensor comes from translation symmetry applied to Noether’s theorem.

However, in general relativity, we tend to define the stress-energy tensor from a variation in the metric. How are these two viewpoints related?

The “usual” method produces non-unique stress-energy tensors, while the GR version seems to be unique. Why is this?

Finally, can we do a similar thing to derive other conserved quantities in a unique way? Can we get the angular momentum tensor from some metric variation?

How about more general conserved quantities that come from symmetries other than coordinate transformations?