Fate has decided in The Legend of Zelda that Link is destined to repeatedly save Hyrule from the machinations of Ganon and other villains. Players are able to change the hero's name in their save file, but the canonical name of Link has become iconic. Link is an apparently not an uncommon name in Hyrule, since it is passed down through the ages, and is even used for people and animals other than Hylians.
Link comes face-to-face with many doppelgängers throughout the series, such as Dark Link and Shadow Links. These usually appear to be some form of manifestation of evil, or the corrupted spirit of a defeated Link from another part of the wholly confusing Zelda timeline. They can hardly be called characters, since they don't really have any - well, characterization - besides being a villain players need to defeat.
For a while, other characters named Link appearing in The Legend of Zelda games seemed to be a recurring theme. Once the series made the jump to the Nintendo 64, four straight 3D Legend of Zelda titles featured a side character also named Link.
Other Zelda Characters Named Link - Link the Goron
Although Link can transform into a Goron in Majora's Mask by using the Goron Mask, there are other Gorons named Link in Ocarina of Time and Majora's Mask. After Link clears the violent monsters from Dodongo's Cavern the leader of the Gorons, Darunia, names his son after Link. Link the Goron is then encountered when Link returns to Goron City as an adult.
Link the Goron in Majora's Mask is a separate character than the one in Ocarina of Time (and is much better dressed). He only appears briefly in Majora's Mask, having a reservation at the Stock Pot Inn, which Hylian Link is able to swindle from the Goron because of their shared name.
Other Zelda Characters Named Link - Link the Pig
Pigs apparently thrived in the post-deluge world of The Wind Waker, because they can be seen in many locations, including Link's home on Outset Island. Before leaving on his grand adventure across the Great Sea with the King of Red Lions, Link can catch pet pigs for his neighbors Rose and Abe. When Link returns to Outset Island later in the game, he finds an incredibly large pig named after him living in the pen beside Rose and Abe's house.
Other Zelda Characters Named Link - Link the Cat
Link isn't the only Link living in Twilight Princess' Ordon Village. Shop owner Sera has a cat named Link that has gotten in trouble for stealing a fish meant for Sera and her family. Cat Link can be found by the pond, trying to catch fish of his own out of the water. In order to get Cat Link back to a distraught Sera so she can run her shop, Hylian Link has to catch a couple fish, one of which will be stolen by Cat Link.