Institutionally empowered would have been a better phrase.
While the PO doesn't have specific legislative powers to bring private prosecutions, they did have a significant amount of institutional holdover from when is was a public body.
Specifically, a close working relationship with the police and other bodies with statutory powers that went far beyond any relationship a privately owned body should have. E.g. they were given direct access to the Police National Computer System.
No government organisation (typically the Crown Prosecution Service in England and Wales) was involved in bringing these prosecutions.
The Post Office itself was legislatively empowered to bring private prosecutions of their employees to the state courts of England.
The whole thing is insane.