The Sword of Troy does not feature in any surviving version of the Trojan War myth. It is not part of the Homeric or post-Homeric Greek tradition regarding the story. It is an original creation by David Benioff, the writer of the movie Troy (2004), presumably to offer some kind of physical manifestation of the survival of Troy and its royal line beyond the fall of the city. It is possible that Benioff intended for the sword to create a link to a possible sequel movie dealing with the content of Homer's Odyssey or Vergil's Aeneid. However, if such plans existed, they never came to fruition, and so the Sword of Troy is a dead letter.
Generally, while we don't know the succession practices of the Mycenaean period in which the story of Troy is supposedly set, the Greek rulers of the world sketched by Homer do not hold inherited positions. Their status is not derived from their lineage or any artifacts of kingship, but from their possessions and their deeds. In the Homeric epics, it is stated at some length by heroes on both the Greek and the Trojan side that the authority of "kings" rests entirely on their ability to prove their worthiness to lead. They do this primarily by offering good counsel, being honourable hosts, and performing great feats in battle. Should the "kings" fall short of what is expected of them, they may lose their status; there is nothing to guarantee that it will be passed on to their sons. Indeed, the entire plot of the Odyssey rests on the assumption that Telemachos will not succeed his father to rule Ithaka unless he can prove himself worthier than Penelope's suitors. In this context, the notion of a "Sword of Troy" and its implication of a continuous line of royal rule is slightly absurd; a king's possessions existed to be given away in gestures of reciprocity and generosity that cemented their status at the top of the food chain.
In addition, insofar as it matters that the movie Troy sets itself in the 13th century BC, it is worth noting that the supposedly ancestral and ancient Sword of Troy is clearly made of steel, which had not yet been invented (this being the Late Bronze Age).