They wouldn’t see what sites you give the tokens to — unless those sites choose to phone home, for some reason.

• You log in to the government site
• You ask for a token to prove your age/gender/whatever
• You copy the token
• You go to the age/gender-restricted site
• You provide the token
• The restricted site asks the government site how to verify any arbitrary token (but doesn’t mention your specific token)
• The restricted site verifies the token