His post resonates very well with my thinking: an identity, even a digital one, is a lot more than "a set of claims made by one digital entity about itself or another digital identity" (from Wikipedia, which probably got it from the Identity Gang Wiki).

He proposes this mental experiment:

One way of sussing this out is to ask: do identity twins have different identities? We would say yes, even when we can't tell them apart.

The "even if we can't tell them apart" is the key phrase here. There is (digital) identity even without claims, and there certainly are different (digital) identities with the same claims!