Business card for a student?
First of all you mentioned that you already finished your bachelors. In Germany this is normally considered a "berufsqualifizierender Abschluss" which means that with your B.Sc./B.Eng. (at least "on paper") you are considered a trained professional capable of doing a job related to your field of study. So nothing "pretentious" about that.

When I was in the same stage (B.Sc. finished, M.Sc. in progress, also in Germany) I also had no business card and in hindsight it was bad to not have any. I was very often asked for a card at conferences or after presenting our work to potential or current industry partners. One older guy at a conference in Germany even got a little "angry" at me when i couldn´t give him contact information in form of a card because he thought it was unprofessional for students not to have one.

As suggested in an other answer just keep it simple (just your name, kind of your bachelors, department and contact info). Maybe your university even has a template or guidelines for it.
And if you worry to be seen as pretentious, just don´t spread them like flyers but wait for people to ask for them.