Yes. The date of opening must be noted on the bottle.
In Germany, hand and skin disinfectants are medicinal products. According to § 11(1), Sentence 1, No. 6 b of the German Medicines Act, the package insert (or, in the case of original packages, the label on the back) must contain the "information on shelf life after opening the container". The corresponding duration is tested and approved by the Federal Office for Drugs and Medical Devices. Usually, the use-by date is much shorter than the expiry date specified by the manufacturer for the unopened pack. For this reason, the user must note the date of opening on the bottle. Hand disinfectants can be used up to 12 months after opening. If they are used in dispensers, this period varies from 6 to 12 months.