Okay, so I'm basically just shattering the idea that the Hero's shade is the Hero of Time (and my main reason to do so is that from a creator's point of view, it doesn't make much sense to wanna show an old character and make him look nothing like he used to). But what good is that without offering another, maybe even better explanation? Well, so here's my idea:
The Hero's shade was the one who should have stopped Ganondorf before he was banished to the Twilight Realm. If he had succeeded and banned/killed Ganondorf or whatever, the Sages would not have had to interfere and risk the life of one of them. But for some strange reason, that Hero did not accomplish his mission - he failed and died.
[...]
My personal claim: The Hero's shade is an ancestor of Link, like his father or grandfather or something like that. He was supposed to stop Ganondorf, but he failed. So he has a similar role as Link's uncle in ALttP. He has failed with his mission, but he helps Link on his quest to make up for his failure. In his life, he accepted the destiny of a Hero (= meant to stop Ganondorf and bring peace), but he wasn't ready for some reason or something like that. He wasn't the right choice, because the next legendary Hero in line wasn't him, but Twilight Princess-Link.