I'm working in a scrum team (about 8 people) without a scrum master
and we have one team member that has a non-development role (not PO,
This is, by definition, not Scrum. The "rules" to Scrum are laid out in the Scrum Guide. What you describe breaks a few of those rules, and if you aren't following the rules, you shouldn't call what you do Scrum. Note that it's OK to not do Scrum - it's just confusing and can lead to you getting advice that's less than helpful.
This person doesn't really fit in, he should actually belong to a
different department, but someone higher in the hierarchy decided
that's where he belongs.
Not only does this break the rules of Scrum, but even more fundamental values and principles of Agile Software Development. Teams should be self-organizing - people should not be forced onto a team where they cannot contribute.
Ideally he should not attend the daily but I think it would be also
not good to exclude him, then he would be in a team without actually
being a team member.
It sounds like he's not a team member. The team is a group of people working toward common goals and objectives. This individual is working toward a different set of goals and objectives. Perhaps it makes sense from a people management perspective to group him with the others on this team, but it doesn't make sense from a project management or team organization perspective.
My question is, does anyone have a valueable experience how to make
him contribute to the daily in a meaningful manner? Just make him say
as little as possible? He talks about stuff that nobody else is
familiar with in the daily.
If he's not contributing to the team, he shouldn't come. Give the guy an extra hour a week to do whatever he needs to do. If he does end up working with the team, then he can come for coordination and collaboration. Plus, you free up the team from needing to process information that is not relevant to their work, giving them more time.