it is propaganda, the empire tries to pass off the amulet as something only the septims can wear, that is a blatent lie. empress alessia was the first leader of the empire and it was from her soul the amulet of kings was made, her son, belharza, was the second ruler of the empire and, one can assume, the first the wear the amulet of kings. now tiber is a gneral under cuhlecain, when cuhlecain was assasinated tiber went on to become emperor of the third empire. it was not untill now that the septims have worn the amulet. it took some searching to find his name but belharza is my proof that the septims are not the only ones to wear the amulet
it s a blatent lie and imperial propagada that only the septims could wear the amuelt.
I still think Treng is right since that's exactly what happened in Oblivion.
Alessia made the pact with Akatosh, received the dragon blood, and had her soul taken into the amulet of kings. Her heirs wear it, and according to legend Reman Cyrodiil is born with it, his mother being the ghost of Alessia. It was at that time when the Amulet became a legitimate symbol of right to rule. During the Reman Dynasty, Talos was told by the Greybeards (led by the second born of Akatosh) that he would rule Tamriel. So Akatosh endorsed the Septims, and with his blessing (and dragon blood), went south to become emperor.
You now have that same dragon blood, known to be held only by royalty.
Also, Delphine is a lying, manipulative ***** when she says Tiber Septim wouldn't have founded the empire if he listened to the Greybeards.
Edit: Since it was kind of hard to prove Reman was born from a ghost, he probably started using the Amulet as sort of a blood test, to show he shared blood with Akatosh and was part of Alessia's family - that family being the only one blessed with Akatosh's blood. Tiber Septim wasn't related, but blessed separately.