Okay. I solved it. I actually had the problem of Non-University Email.
Initially, I used the shortend version for the Ruhr-Universität Bochum (rub.de) for my email address. When I checked for this problem the first time, I tried to create a new account with the fully written out domain (ruhr-uni-bochum.de). This wasn't possible and I got the alert that this email address was already in use. So I falsly thought that internally Matlab/MathWorks already linked these addresses.
It turns out, I had to still update the email address in my profil with the "ruhr-uni-bochum.de" version. Then the login via my Uni worked.
However, my inital question still somewhat remains, how I as an University student can make a service request at Customer Service without a valid licence.