Can a Minecraft admin read private (/ msg.) Messages?

Would be loud https://dejure.org/gesetze/TKG/88.html forbidden, right? On forums etc. It is also forbidden. Does that also count for German Minecraft servers?

Yes he is allowed, he is a so-called guardian.

You assume that the players would be underage?

I assume that the server has some rules and you can then also look. In addition, the server owner has his own decide whether he may or not.

So you think that the owner of a server can and can override applicable law?

Yes, he is the leader. He is allowed to set rules. And Minecraft would hardly have made the function that he could not.