Deutsche Boerse AG runs the XETRA system and is a private, publicly traded company. While they are of course overseen by the financial authorities, the German stock exchanges are private institutions.
In English, a public company is a company whose shares can be freely traded by the public.
In German, a public company is a company that is owned to the government. Companies that are listed on the stock market are called "kotiert".
This could hint at a cultural difference, with the Germans more strongly identifying with the government ("if it belongs to the government, it belongs to everyone"), whereas the English speaking countries might tend to identify more strongly with the economy ("if everyone owns or at least could own some shares of it, it belongs to everyone").
At least public sector and public vs private in the context of privatisation are used in the UK also.
Also public and private schools (in the US.)
Also in the UK the "Public Sector" is government organisations.
The "private sector" is for-profit/comercial organisations, many of which are publicly traded comapanies, which is normally shortened to simply "public companies". So you could be a public company in the private sector that provides services to a public school, which is part of the private education sector.
Language is fun.