How can we be sure that the person operating the wallet is the same person whose PID is in the wallet? For example, can a child operates the wallet on behalf of an elder parent?

We will set up a prerequisite that the person able to unlock the device is assumed to be in control of the device.
This is the model followed by other apps like banking apps. If you are in control of the wallet, you are assumed to be the owner of it.
Regarding granting "the power of attorney", to another individual like a family member, attestations will exist that grant it for use with the EU Digital Identity Wallet.

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How can we be sure that the person operating the wallet is the same person whose PID is in the wallet? For example, can a child operates the wallet on behalf of an elder parent?