It's not an EU project, it's individual governments making long-term infrastructure decisions. Just like the French gov chose matrix for internal communication. Sadly there has been no national move to Linux as desktop OS for staff yet, but quite a few lower level initiatives. Those lower level ones obviously have less resources and more fluctuation, but shifting 300k federal employees to a new storage provider will not be a frequent decision.
NextCloud was an independent company well before it did anything government related.
The only thing that's happening here is they appear to have struck a huge contract with a few EU countries to build internal clouds.
I believe NextCloud is open source, and a fork of owncloud from the original owncloud developer who wasn't happy with the direction it was taking, or something like that. Also, they are a profitable company, and the product is open source, so this only looks like a boost to their business model and the open source product it's based on.
(Edit: removed incorrect information.)