These two characters may look quite similar, but they can't be siblings for reasons outlined above.
The Legend of Zelda chronology may be a bit confused and convoluted, but it's clearly not the same Link and Zelda in every game (unlike the Mario games which feature the same Mario and Peach ad infinitum): Ocarina of Time and Majora's Mask feature the first Link and the first Zelda, A Link to the Past and Link's Awakening are set at least a full generation later with a new Link and Zelda, the original two NES games are set generations after that, and the others fall in between somehow. So, each Link is either a descendant of the original Link and/or is his reincarnation. Similarly, each Zelda is a descendant of the original Zelda and/or is her reincarnation (given that they're all the princess of Hyrule, then I'll go a step further and say that the Zeldas must be related). The question now is to ask if the Link and Zelda bloodlines crossed at some point. Given that the original NES games suggested a love interest between that pair, I would certainly hope not because that pair would then be blood relatives! So, unless the Hyrule aristocracy is taking a rather gross page from the Borgias or the Lannisters: I highly doubt that any of the Links and Zeldas are siblings.
The uncle from A Link to Tthe Past was probably going to say something along the lines of "destiny," which would reference the original Link and Zelda and their banishing of Ganon to the Dark World (Ganon having cursed the descendants of Link, Zelda, and all of the other sages as he was banished).