This afternoon I went into my Post Office, which is the Sectional Center (Main Post Office) for Northern Virginia (and just outside Washington DC). I checked with two window clerks and the shift supervisor. The concensous was it is an interesting rumor, but only a rumor. There is no truth to ANY requiring of ID when mailing a package.
However, for several years the Post Office has had a rule, at the request of the FAA, that no package over one pound could be sent via Air Mail (including Priority Mail) unless it was presented at a window (even if proper postage was on it).