So just now I just created a new folder and moved the blendfile and the normal map (to send together as a zip) to this new folder. Then I reopened the .blend file to reload the normal map with the new image path and to see if the problem is still there and now it is working for some reason. So I am pretty sure it was a bug and restarting blender or moving the files seems to have resolved it. Though in the end I am still not sure what exactly caused this or what made it work again (I literally changed nothing apart from moving the file and reopening Blender). Thank you a lot for your time and help.
You girls/guys just saved my life. First normal map bake since the anvil tutorial and having exactly the same issues in blender 3.0. Normal maps are being displayed correctly in eevee but not in cycles. My shading was even weirder. I just moved all of the files and the normal map to an new folder and voilà! Everything back to normal (snicker).
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Thank you, it took me some hours but restart of Blender solved it!