Most people involved in federal agencies don't get purged and replaced at every election; they have long careers in their chosen agency, regardless of who's in the White House. So it's easy to find someone who's been there through a few administrations and say "this guy has consistently proposed this idea under multiple administrations", but present it as "this idea dates to the administration I want to falsely paint as identical to another one".
The related trick is the West Wing bit with the reporter who keeps asking the press secretary to give an answer on whether the President has considered calling a lame-duck session of Congress: the reporter knows, and the press secretary knows, that if she asks the President whether it's been considered then it'll be true that the topic came up in a discussion the President was part of, and the reporter will be able to spin that to claim "PRESIDENT CONSIDERING LAME-DUCK SESSION" without an outright lie being involved.
Most people involved in federal agencies don't get purged and replaced at every election; they have long careers in their chosen agency, regardless of who's in the White House. So it's easy to find someone who's been there through a few administrations and say "this guy has consistently proposed this idea under multiple administrations", but present it as "this idea dates to the administration I want to falsely paint as identical to another one".
The related trick is the West Wing bit with the reporter who keeps asking the press secretary to give an answer on whether the President has considered calling a lame-duck session of Congress: the reporter knows, and the press secretary knows, that if she asks the President whether it's been considered then it'll be true that the topic came up in a discussion the President was part of, and the reporter will be able to spin that to claim "PRESIDENT CONSIDERING LAME-DUCK SESSION" without an outright lie being involved.