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?
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.