Not quite. The PO will not replace a damaged box or remove it if it is no longer in use. However it is protected from vandalism and from use by other than PO employees.
The government considers mailboxes their personal receptacle to be stuffed with whatever they deem appropriate. That is, any material bearing a PO stamp. All other material is contraband and its purveyors are subject to penalty.
Deliberate destruction of mailboxes is similarly subject to penalty. Removal of your personal box is a gray area. There is, as yet, no law requiring a citizen to receive material from the PO.
This is a federal, not a local issue. Hence, the stickers are a violation of federal law unless placed there by a PO employee or their representative. Proceed with extreme care.
You ALSO can place other boxes out there for any other sort of reason you can think of.
If you don't want mail, don't put a box out there but don't expect the post office to save it up for you. Those "information" notices your creditors send to you will be returned and they'll stop mailing you the notices ~ and they'll probably cut off your account as well, but that's not the post office at work.