It's true it could be a plausible assumption that Link and Zelda have a similar, or even related background. However, I definitely don't think they are related as siblings.
For one, Zelda lives with her father, the king, and Link doesn't have any parents. Evidently his mother died after bringing him to the Kokiri Forest, and we can assume that Link's father died in a war, supposedly the war that takes place before Ocaina of Time.
If Link and Zelda were brother and sister, why would Link be brought to the forest to live and Zelda remain with her father in Hyrule castle?
On the contrary, you could argue that this is because Link was seperated because of someone knowing his destiny, although I don't think that is very likely.