The problem comes when company A wishes to ensure that its interactions with company B involve individual X, and not some other individual Y or Z.
It's well and good for a given organisation to have its own internal identifier. And virtually all do -- that's what your account number, or customer ID, or payment number, etc., are. That works fine for internal operations.[1]
But it doesn't address the inter-organisational problem.
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Notes:
1. Well, sometimes. Actually the hijinks that can happen within an organisation with identifiers are ... numerous.