a) I spent some time looking for this today, with no success. In a lot of years reading about travel and documentation and visas, I've never read of the DoS issuing such certificates. No one else has chimed in with a portal or reporting prior success in obtaining such affirmation from the State Department.
While it's not logically possible to prove a negative, the lack of any positive evidence, along with a plausible explanation (see below), convinces me that the State Department does not issue them.
b) Another country's consular staff thinks the US State Department does issue such documents. This isn't surprising. The central government of some or many other countries does indeed issue such documents. If you're from one of those countries, it's reasonable to assume that every country does. I'm sure whoever on the consular staff said the DoS does issue such certificates will be unable to point you at exactly how one applies. The FBI is close (central government), but no cigar (Dept of Justice, not the Dept of State).
TL;DR: The US Department of State doesn't issue such verifications or certificates.