The only entity able to connect you in this case is the identity verification third party. The premise is that a government-backed identification system is more secure than a rando private company.

Private company asks government “hey is this user real and unique”, government replies “yes”. Private webiste does not need to know your ID. No identifying element needs to be transmitted by the government.

Of course some private companies will need more, and in that case the user, you, can grant them access to data, much like the current authentication systems using Google accounts & co.

In which case the flow would be:

• Rando insecure company asks government "is this user real and unique? I need their name"
• Government website asks you "this rando company wants to know your name"
• You accept
• Goverbment replies to rando insecure conpany “yes, user real, name is X”

That’s how it should be.