Danse isn't a copy of a "real Danse", he's an escaped synth that the railroad memory wiped and smuggled to the capital wasteland, everything after that is stuff he actually lived through, including joining the brotherhood and rising up the ranks.
As The Appalachian mentioned, Nick has a copy of the "real" Nick's memories and personality, but for whatever reason, the Gen 3 infiltrators don't seem to have them, as the militia in Goodneighbor successfully uncover and kill an infiltrator because he stopped being an alcoholic and cheating on his wife out of nowhere, the same happens with Roger Warwick and McDonough.
This is likely because the Institute makes their infiltrator doubles before they even abduct their victims, as seen with in the Synth Art vs Art random encounter and the memory scan likely can't be done at a range. So while they may have the technology, they aren't using it for whatever reason.