Yep,proof of possession - the ID contains a public key, you own the matching private key. You send them the ID with a matching assertion ("This is meant for cheapboozeforstudents.com"), signed by your private key. The website can't in turn generate an assertion for statestudentaid.gov because the whole envelope (ID, assertion, signature) is uniquely tied to them.

It does rely on statestudentaid.gov actually validating said assertion, of course, but that's only like... The second most common screw up with JWTs and related tokens.