With Zelda I'm fairly certain it's canon that by tradition each next daughter in line has the name passed down to her. Obviously, with the necessary time gaps between games, there would have to be more Zeldas not involved in some sort of calamity or conflict depicted in the series.
Just wanted to address that first for the sake of comparison. In the case of Link, we don't know for sure that the name is passed down generation to generation. If it is then likewise there must be Links who never become heroes. It's less certain, however, that the Links of the different games are even related to one another. In The Wind Waker they say outright that Link is not a literal descendant of the Hero of Time (although you might say that's a special case what with the split timeline).
There's also the element that the name Link is supposed to be more or less a filler name. In other words, Link is the default name for the hero of every story, but by virtue of both the name's meaning and the fact that you can choose his name in almost every game, what each Link is actually named is somewhat ambiguous, meaning there's no solid canon name for any Links (except perhaps in The Legend of Zelda and in Four Swords Adventures). Link's name is whatever you name him; it's possible that each hero is in fact named Link, while it's also equally possible that each hero has a different name entirely.