The way identity wallets work:
The government issues an eID to your wallet. 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. The government signs individual attributes combined with the public key with the government private key. The government certificate containing the public key is, well, public.
One of the attributes is ‘over_18’ (In the EU eID scheme countries can add other over_XX attributes if they want, but over_18 is mandatory).
When a website wants to requests attributes, in this case the over_18 attribute, they send a request to the user’s wallet app, including a challenge. The wallet sends back a package including the government-signed attribute, which contains the device public key and the over_18 attribute plus a response to the challenge (proving the credential didn’t get transferred).
The website only sees the ‘over_18’ attribute, which is backed by the government signature. They don’t see any other attributes (the wallet app shows in advance which attributes you are sharing). The government never sees which website wants to know if you’re 18+.
Of course this is all a bit simplified, check OIDC4VCI and OIDC4VP for details.
The only real issue is the wallet app and device binding. Because a compromised device could allow credentials to be transferred some form of attestation of device and wallet app is required. In practice this means no rooted/jailbroken phones.