There is also the German state of Lower Saxony holding 12% of Volkswagen, and of course lots of countries fully or partially owning technically private organiza

There is also the German state of Lower Saxony holding 12% of Volkswagen, and of course lots of countries fully or partially owning technically private organizations in infrastructure, transportation and communication. So "generally" is doing some lifting here. It's not that this never happens, but it is the exception. It would be highly unusual for the EU to just directly invest in companies like that.