You cannot turn a ZKP into being secretely not ZKP without significant effort though.

Take the following example protocol:

1. Social media app sends you a token to verify.
2. You append a private secrete string to the token and hash it with a known, collision resistant hash function.
3. You send the hash to the government's server and request an "18+" signature. The signature should correspond to a public key.
4. You send the signature back to the social media app, including the secret you used.
5. The social media calculates the hash of its token + your secret and then checks whether the governmen't signature is valid with that value.

The government will not know which social media site was used, the social media site will not discover anything about your identity beyond a binary "is above 18 years old" statement. This is because you control all communication.

To discover anything else, they would BOTH have to collude in some significant way. They can only do so in step 5, by having the social media app send the value you gave it to the government. Maybe there exists a protocol that you control that works against this threat as well, I'm not sure.

But if they collude in step 5 - what prevents the social media company from sending all information it has about you to the government already all the time, even without age verification? Like IP addresses, phone number, access time etc. If the government further controls all the ISP servers and log which traffic from where goes where, it could certainly identify you already.