You're speaking about an
ITIN as opposed to a Social Security Number? The IRS keeps records of ITIN's, just as the Social Security Administration keeps track of Social Security Numbers. Not everybody qualifies for an ITIN, and employers are going to want a SSN.
Credit bureaus will attempt to match up the new number with the old account. If you change your name through a court, there's a public record of your name change. A creditor who wants to find you will eventually find you.