I think when my mom got a PO Box they wanted hers, and they checked mine too, so I could pick up mail for her. I never read the whole application though:
See pages 4-6 of the application here:
http://www.usps.com/forms/_pdf/ps1093.pdfexcerpts
Updating Information. The information on your application (PS Form
1093) must always be current. As soon as any information changes (such
as address, telephone number, etc.), you are responsible for updating the
form. Failure to update the application may result in a termination of
service. We keep the form on file at the office where you use the service.
Use of Information. The U.S. Postal Service® is authorized by 39
U.S.C. 403 and 404 to collect the information on PS Form 1093. We use
the information in providing Post Office box or caller service to the
applicant, but we may disclose it:
To anyone authorized by law to serve judicial process.
To a government agency, in performance of its duties.
To a congressional office, if the boxholder requests it.
In response to a subpoena or court order.
When it pertains to a legal proceeding that involves the Postal Service.
The complete Privacy Act Statement is on the back of the application card.
on the application:
9. ID and Physical Address Verified by(Initials)
11. Two types of identification are required. One must
contain a photograph of the adressee(s). Social
Security cards, credit cards, and birth certificates are
unacceptable as identification. Write in identifiying
information. Subject to verification.
back of application:
Privacy Act Statement: Your information will be used to provide Post Office™ box or caller
services. Collection is authorized by 39 USC 401, 403, & 404.
Providing the information is voluntary, but if not provided, we will be unable to provide this
service to you. We do not disclose your information to third parties without your consent,
except to facilitate the transaction, to act on your behalf or request, or as legally required.
This includes the following limited circumstances: to a congressional office on your behalf; to
financial entities regarding financial transaction issues; to a USPS auditor; to entities,
including law enforcement, as required by law or in legal proceedings; to contractors and
other entities aiding us to fulfill the service (service providers); to process servers; to
domestic government agencies if needed as part of their duties; and to a foreign
government agency for violations and alleged violations of law. Information concerning an
individual boxholder who has filed a protective court order with the postmaster will not be
disclosed except pursuant to court order.