Welcome to the community, @Dietwald_Claus! :grinning_face_with_big_eyes: Sorry that this wasn’t answered sooner. What you’re seeing with the form behavior can’t be changed. When setting up the form, you see all of the related fields for linked records because you’re the form creator. When users open the form, though, all they see is what’s in the primary field for linked records. The only way to natively give the users more to choose from is to use a formula field as your primary field in any table that you want to link to, and pull in data from other relevant fields that the users might wish to search through.
If that won’t work for your use case, you’ll probably need to consider a third-party form tool that may let you give users additional searching options. I know that On2Air Forms by @openside, which ties into JotForm, can let you do more robust searching. I don’t have experience with other form tools, so I can’t speak to their specific abilities to address this issue.