The ID is signed by the government and linked to the device to prevent transferring the credential. A public/private key-pair is generated by the secure enclave in your phone, the public key along with proof of possession of the private key is included in the request for the government eID.

IMO, there are two other issues that need to be solved. The major one is that there should be some way to do attestation of devices that are not Google-certified Android or iOS. If this does not happen, the smartphone duopoly is permanently entrenched and not a fair/free market anymore. There is no way to use a smartphone without basically losing your privacy to Google/Apple and given the increasing importance of online services it's becoming increasingly impossible to live without a smartphone.

It was very disheartening that the EU reference implementation was rolled out with only Play Integrity and Apple's counterpart. IMO, this should have been solved before the reference implementation was rolled out to member countries, because many of them won't bother to go beyond that [1]. It is also completely counterproductive when it comes to EU tech sovereignty. There is a group of pioneers that are growing the sovereign ecosystems and then you cut them off.

The second, perhaps lesser, problem is that the security story is not super strong, because most Android phones do not even have a secure enclave (outside Pixel and Samsung flagships/A5x, there are very few). Instead they rely on TrustZone etc. which are regularly targeted by side-channel attacks, etc. Ironically, GrapheneOS is cut off from most of these systems (because Google Play Integrity), while it actually requires a secure enclave and is more secure than... well I guess every other smartphone.

[1] There is some hope, e.g. the developers of the Dutch identity wallet acknowledge the issue and are open to supporting alternative systems.