I have a current military ID (CAC) and it mentions nothing about what to do if found. Not something I’d lose though, what with name, birthdate, and social on it :/
I can't honestly remember. I do remember having an ID that said to do that if found, but I can't say for certain if it was my Military ID (US Air Force), or something else I had at the time. It may have been my driver's license from Idaho. I just remember the statement of dropping it in the mail if found.