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Can You Prove Two Particles Are Identical?

I think Eliezer wasn't specific enough what he meant by "indistinguishable". In the QM and thermodynamics/chemistry sense the particles are indistinguishable iff they are completely described by their quantum states - which are just finite sets of numbers. The properties of statistics change a lot when moving from distinc to non-distinct particles - think of permutations versus combinations.

The Quantum Arena

But wait, wouldn't that still imply there's no unique way to slice something into a positional configuration space?

The space-time interval is a conserved quantity under Lorentz transforms. The position or time interval separatedly are not conserved quantities. Since the spacetime intervals are conserved, the configuration space is unique and consistent.