Official Link: Downward Thrust, Mother Fucker!!

Translation:

→French: Link et Zelda ne sont de la même famille que dans ta tête.

English: Link and Zelda are from the same family only in your mind.

Actually I think they're siblings in Zelda II (and if I recall Zelda II is a direct sequel to Zelda I).

→ I don't remember that being mentioned. Plus they kiss at the end, so... however, it is implied that they are in a Link the the Past, by Link's uncle speech before he dies.

In the Japanese version, he says "Zelda is your destiny." (Only, you know, in Japanese.)

→ Oh thanks! Well that makes a little bit more sense, then.

why do they kiss at the end of zelda II!?
lol. i doubt miyamoto had that in his vision...
still ive always wondered....
in every zelda game: theres a different link. of different generations. like, a looonng family history.
JUST WHO DOES EACH LINK MARRY TO REPRODUCE?! that question boggles my mind....
ehhh... minor gripe i guess. still, link is awesome in this game! (isnt every character in this game?!)

The Zeldas are all descendants of each other, but I don't think the Links necessarily are. It's entirely possible none of the Links are related to each other, or to any of the Zeldas.