Take, for example, OpenText. [1]
Nothing about it is free, or open source code, or in any other sense publicly accessible. [2]
I don't know if they operate in the EU, but I presume any application for a trademark would be rejected.
[1] https://en.wikipedia.org/wiki/OpenText
[2] It's "Open" as it turns paper documents into more easily accessible digital formats. But not public in any way.
[1] Actually it originally did in 1833. It doesn't today. But that’s still the name.