How can we be sure that the person operating the wallet is the same person whose Digital ID is in the wallet? For example, can a child operate the wallet on behalf of their elderly parent?

The wallet will assure through a variety of security features (e.g. PIN codes, biometrics) that it is usable only by the person to whom it has been issued. It will be illegal under national law to use the wallet of another person unless this person has given explicit consent.