Hello Markus

It shouldn't behave this way. If you configured the column to be a Foreign Key column, the data you see in the column should still be found through filters and search.

As you can see in my example. This is the setup of the "TEST" column:

And, search works fine:

So does the filter:

If there's something else you were referring to, please explain in a bit more detail, but if this doesn't work for you, please provide me a temporary WP-admin login for your site where this happens, so we could log in and take a look ‘from the inside’ as that’s the most efficient way to see and resolve the issue. We do not interfere with any data or anything else except for the plugin (in case that’s a production version of the site), and of course we do not provide login data to third party. 

You can write credentials here just check Private Reply so nobody can see them except us.