I once did work for the Post Office Investigations Division. I was "hired help" - I was employed by a hardware vendor. The people I was working with were pleasant, decent and clever people; but they explained to me that the organization was secretive, and operated rather like a cross between MI5 and Special Branch. That's not similar to a normal private prosecution. The UK now has a Crown Prosecution Service; the police pass a completed investigation to the CPS, and the CPS decides whether charges are justified. There are laws restricting what police investigators are allowed to do and say. None of that applies to the Post Office Investigations Division (who apparently employ their own prosecuting counsel).

I think the Wikipedia article you linked explains that the Post Office's power to investigate, charge and prosecute dates back to the 17thC. So yes, "anachronism" is right.