Rivals

[Dark Link]

Dark Link vs. Link - Do I even need to explain why this might end up being a thing?

Dark Link vs. Ganondorf - Since the scant memories/recollections that he has come from Link's perspective, you can imagine that DL won't react kindly to GD once they encounter one another and might even pull a Link and just attack the man on sight. Regardless of how their relationship ends up developing, I imagine that at least one fight between them is unavoidable.

Dark Link vs. Gilgamesh - Actually, on its own, I don't think a rivalry between these two is particularly likely. DL has no personal reason to be hostile towards Gil and Gil would probably consider it beneath him to deal with someone like DL most of the time. That said, if they did end up clashing for whatever reason (which isn't as unlikely as it may appear, given that DL is an Empire agent), I could see Gil making it a point to be as belittling and hostile towards DL as possible and thus escalating their relationship into a sort of rivalry. Plus, who else would be more fitting to act as the Berserker in the Omniverse (even though DL is a polar opposite of Berserker in terms of personality)?