The government already knows my name and where I live, so if I trust them or not isn’t important.
To alleviate the issue with copied cards, cryptographic signatures and certificates can be used. The certificate is on your card and it’s signed by the government. Your certificate contains a private and a public key. The private key never leaves the card, only the public key. The public key is signed by the government.
So if a service wants to ensure your identity, they will give you a cryptographic challenge. That is then signed using your private key on your card. The service then can verify your signature by using your public key. The service can then verify your public key by contacting a government server. That way you can prove to anyone that you’re in possession of the physical ID card. The private key on the card can be further protected by a pass phrase.
So if someone wants to steal your identity, they would need your physical card and your passphrase.
If your physical card and passphrase gets stolen, you report it to the government and they revoke their signature of your public key. So if a service wants to verify your identity, the government server will reply that the ID is not valid.
This is how SSL and public private key cryptography works in general. The issuing authorities don’t have to be governments.