McCoy doesn't mention anything about blood composition. Besides, Original Khan and nuKhan are not the same people.
He doesn't have to. The human healing process centers around blood. Khan was seconds away from death in "Space Seed", that he could recover quickly and completely without McCoy lifting finger means he has "magic blood".
Original Khan and nuKhan are the same people. Khan comes from before the timeline split in 2233.
Not every little thing needs a piece of dialog to fully "explain" it. The viewer should be able to connect the dots on their own.
Berman Trek Magic Blood
The Doctor pronounces Neelix dead, but Seven proposes a revival procedure using nanoprobes from her bloodstream. After discussing it with the Doctor and Captain Janeway, Janeway orders the Doctor to proceed, even though Neelix has been dead for 18 hours. Seven quickly modifies and administers the nanoprobes, they proceed, and after some touch and go, Neelix is revived. He has no recollection of what happened and where he has been and wonders why he is in sickbay.
Gee that sounds familiar. Like I saw it in a movie recently.
We also have Nomad raising Scotty from the dead in "The Changeling".