We had this problem with a Paperwhite 1. The notebook with the details was lost. Even an old Kindle Keyboard 3 (FW 3.something) has only the user name.
We had all the email addresses allegedly ever used except maybe one and that didn't help. The one possible unchecked email was using a domain long gone.
As there was a new Kobo Sage rather than a new Kindle we just re-registered it to a new Amazon account, which worked.
I can't see why Amazon would tell you because how can you prove it's your account? Anyone could obtain an old Kindle that's still registered.
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