I can tell you what is a public bank in Brazil: it is a bank almost equal to a commercial bank. The differences are: the directors are chosen by the government, they have to spend a little more money in social programs, they have absolute guarantee from the government that if something bad happens the clients will be protected by the government and, of course, their directories are used by government as a guarantee of fidelity from the politics to the government.
I am interested about the final thing you said. Friend, what did you mean? "guarantee of fidelity"
@ronaldoavelino do you know people with first-hand experience with Caixa?
(https://en.wikipedia.org/wiki/Caixa_Econ%C3%B4mica_Federal_
That means that the government distributes directories/employment in the Bank to some Political Parties so that the Parties vote with the Government in Congress or Senate. There is a saying: It is giving that you receive. Unfortunately that is Brazilian politics...
About Caixa, I was a client a few years ago, but I don't know anybody that reached or had access to the high levels of power.
Caixa Econômica Federal
Caixa Econômica Federal (Portuguese pronunciation: [ˈkajʃɐ ekoˈnõmikɐ fedeˈɾaw], Federal Savings Bank), also referred to as Caixa or CEF, is a Brazilian bank headquartered in the nation's capital, Brasília. It is the largest 100% government-owned financial institution in Latin America. It is the fourth largest bank in Brazil by assets and one of the largest in Latin America.