It is run as a private company despite being state owned. Some EU regulations require that all European train networks compete like commercial entities which ads limitations to financing the DB and creates a lot of bad incentives. Like letting infrastructure rot, because complete renewals and replacements are paid by the government while upkeep isn't.

GP did say that it got privatized, not that its a private company. Privatization has a range of meanings, which also include things that do not end up in a privately-owned company.