We have this in the US as well, "Registered Mail". It's only used for things that are very sensitive/important since it's a huge pain in the butt, both sender and recipient require verification of identity.
“Registered Mail” in the US does not require any form of verification of identity. All it does is provide extra insurance and delivery confirmation. “Certified Mail” doesn’t require identity verification either, so I’m not sure what service you are thinking of, but I’m near positive nothing of that sort exists with USPS.
Specify the person who can sign for and receive your item. Must be purchased in combination with another extra service as follows: Certified Mail, COD, Insured Mail (over $500), Registered Mail, or Signature Confirmation.
Restricted delivery doesn’t do identity verification. I can say I’m John Doe and sign as John Doe and receive the package (I’ve received restricted delivery packages before and the carrier in multiple different states never asked for ID). I’ve also sent restricted delivery mail before and never had my identity checked as OP claimed.
Edit:
Per Stamps.com [0], the USPS “may” require ID on delivery, but again, in my experience, I’ve never been asked once.
https://www.usps.com/ship/insurance-extra-services.htm