Can I somehow use Focus on my iPad without automatically having it turned on on the iPhone as well?

Yes.

Settings > Focus > Share Across Devices - set to OFF

You are very welcome. I hope this resolves your problem.

The iPad User Guide is a rich source of frequently overlooked information. The Guide is available both online using a web browser (such as Safari), or as an Apple Books download:

iPad User Guide - Apple Support

https://books.apple.com/book/id1567104892