They might not know the list of sites you visit right away in the same way they could by contacting your ISP when you are not using a proxy, but that wasn't my point.
My point is that they can check with a specific site that uses this verification method and see if you have an account on that site, and if you do, which account in particular. And in a way that is much more directly linked to you personally than an IP address (which might be linked to the household/internet access you're using but not necessarily to your name personally).
So in this situation they can indeed know if you use any one particular site that they choose to target, as long as that site is requiring you to provide them with a document, regardless of how many layers of proxies you (or the site) choose to be under.
I'm not sure what you mean by "the site that’s requesting this", the site does not need to request anything from the government, they just need to have previously agreed on a "secret" mathematical verification method that works for every document. The digital equivalent of a stamp/signature.