Do they not already have the equivalent of a US social security card? (For the employment eligibility, not the program benefits.) Is this something much different from that?
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The US tried that back when Social Security Numbers were introduced. It specifically said it was for tax-purposes (a context where it might've been adequately-secure) and not to be used for anything else.
Yet without any actual penalties against "other places", it got misused everywhere by companies trying to save a buck on primary-key choice and authenticating people.