In dark matches before some big cards, Link would take his cousin to the ring to be a manager type character.
Due to how similar they look, and one time when the cousin was helping out Link in battle, they named him Link Jr.
When he appears now, he's also just Link. Some people have started to refer to the original as "Link Classic" now.
Its has more to do with the deal of the promoter owns the rights to the personality of Link even though "Link Classic" left Legend of Zelda to work outside the company. And was replaced by the slim, trimmed, stacked, and, jacked character who appeared in Adventures of Link.