> EU generally doesn't act on the market like that

It absolutely does. The problem is it must then partition the work across the entire continent. America has this problem, too, e.g. the F-35, but the consensus model of EU politics makes it particularly insidious.

Is it just me that think that’s a huge catch-22? The project isn’t eligible for grants because it gets money on the free market, but is then limited by who it can come from or in what amount

Isn't Norway operating under a form of capitalism as well as owning controlling share of Statoil? Why mention China or Russia?

I guess Norway is not a member of EU.

Norway retaining a controlling share in Statoil, now Equinor, is not quite the same as buying a startup or indeed buying any other company.

Statoil was created by the state not bought by it. If anything Statoil is an example of the very opposite idea and that is that Norway has sold shares in a company set up by the state rather than bought shares in an already existing company.

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.

The EU is not a state. There is no possible way of having something like nationalisation at the EU level. So whilst nationalised companies exist in different member-states (almost all of them, I would think), the EU still cannot do that.