How does one say "background check" in German? I'm not sure of the most conventional terminology.
If this is about checking a criminal record the requested document is
Strafregisterauszug (Switzerland, German)
Regulations are complicated. For example in Germany, businesses are generally not allowed to require that document but may prove they are allowed to do so as an exception. For example, banks or child care facilities. They only get information about the things which they may have a legitimate interest in.
If this is about a loans and credit card record, there's the SCHUFA (Schutzgemeinschaft für allgemeine Kreditsicherung) and some other private businesses which bundle information about a possible future payment deficte of persons and businesses. Businesses can pay for a
to get a scoring value about a person or another business. You have to allow that per case. It doesn't list in detail which income/loans/debts someone holds.