I've heard it tangentially, and I think somewhere in the registration database it says to use the confirmation letter as proof and not share your registration number if possible. But the FDA Q&A has this to say:
30.3 Q: Is a facility required to provide its food facility registration number, assigned by FDA when the registration is submitted, to customers or other businesses who request the number. Is a facility prohibited from revealing its registration number?
A: Section 415(a)(5) of the FD&C Act prohibits FDA from publicly disclosing certain registration-related information, including the registration number. However, this prohibition does not prevent a facility itself from disclosing such information. In fact, for imports, a facility will likely need to provide its registration number to any downstream commercial entity who will be submitting prior notice for a food manufactured by the facility. The FD&C Act does not prevent a foreign facility from entering into an agreement with its customers to limit the circumstances in which the facility's registration number may be disclosed to third parties
M.4 FDA’s list of facilities and registration documents are not subject to public disclosure. How do we know that a supplier, for instance, is registered?
Section 415(a)(5) of the FD&C Act provides that certain food facility registration information is not subject to disclosure under FOIA. However, disclosure of such information by the facility itself is not prohibited. FDA expects that generally, foreign suppliers and their customers will resolve this question as part of their agreement to buy and sell food for consumption in the United States.
M.5 Will FDA require the food facility registration number to be displayed as part of a food label?
No. There is no requirement to list on the food label the registration number (or numbers) for the facility (or facilities) associated with manufacturing/processing, packing, or holding the food. FDA actually discourages food facilities from including their registration numbers on the food label to prevent others from using the registration number for improper purposes.
I still am unsure just what exactly these improper purposes would be, other than potentially pretending to be another company when importing?
Ultimately though, since in M.4 FDA clarifies that they'll do nothing to help you confirm whether a facility is actually compliant, them providing an X digit number to you does nothing to verify compliance, so you might as well just get them to say "we're registered" in writing, it's as good as anything else.