You don’t have to read the Personal Identity Card Act for a long time, as Section 1 Paragraph 1 Clause 3 already says: “The identity card holder may not be required to deposit the identity card or give up custody in any other way”. Important “This does not apply to authorities authorized to determine identity or in cases of confiscation and seizure”.
Specifically, if a police officer asks for your ID, you must show it and the officer may take it or confiscate it. But you don’t have to leave the person to the rental company.
In any case, ID or passport are not suitable as a deposit, because according to the law, both documents are the property of the Federal Republic of Germany. Also because of the personal data stored in the chip of the ID card and the possible online use, you should refrain from giving out your ID card. However, the legal regulation does not only apply to ID cards in credit card format.