> My experience with the US tax system is that you need to get approval to get non-profit status,

I think in the majority of European cases you don't need prior approval. The UK is most likely the biggest exception where you can become either a non-profit or a charity. And if you want to become a charity in the UK, then yes there are more hoops to jump thorugh including approval from Charity Commission.

But for Germany for example, you can just go setup a gGmbH which is simply a non-profit/charitable form of the standard GmbH. The only difference is what you put in your articles of association and how you register with the tax authorities, but you don't need prior authorisation for either, you just apply for the status with the tax authorities post-formation.

Whether non-profit or charity you get tax exemption on both in Europe. The only difference is in the donor experience in some places (e.g. in the UK to get a personal tax break you have to donate to a charity, not a non-profit).

But as above, I think the UK is the exception to the rule, I suspect in most EU countries it is closer to being non-profit == charity with no differentiation.