I think I found the solution: After looking into the support information on the German website of ESET I had the idea to first install the test version and to upgrade it to the pro version later and it worked. I didn't even have to make the upgrade because everything happened by itself after choosing the test version.
I've just bought ESET Mobile Security Premium via Google Play Store for my Android device. Now, when I want to start the app I have to answer the same questions (language, country) again and then then I am again asked to buy the product that I've just bought and I can't skip that screen. I can't use the app and I don't want to buy it a second time. What can I do?