Ragna & Jin
Technically from a game (I mean, duh, "Blazblue" IS a fighting game), but since it got an anime, I'll make an exception this time around. Ragna and Jin used to love each other as normal siblings would in the past, but things changed when Saya, their sister, grew sick. Ragna dedicated himself to take care of Saya, and stopped playing with Jin. Jin, being the unreasonable brat that he was, decided to try and kill Ragna when he was possessed by Terumi. He chopped off Ragna's arm and left him to die, and that was the last Ragna saw of Jin, as he disappeared into the NOL.
Meanwhile, Ragna was resurrected by Rachel as a demi-vampire-human badass. He swears revenge on the NOL. He trains under Jubei, then goes on a journey to f**k shit up for the NOL. He meets Jin on multiple occasion, who has become someone completely different. With his twisted love still remaining for Ragna, he believes that his only way to recompensate for what he did as a child...was still, to kill Ragna. Time and time again the 2 butt heads, and in the CP story, even when they were supposed to work together, Jin STILL wanted to get a good cut out of his brother. These 2 just never stop, and I doubt that will ever change.
Shinku & Suigintou
Sisters made by the master doll creator Rozen, the 7 Rozen Maidens were to do battle with one another and take each other's Roza Mystica, so that they can be deemed as the ultimate and perfect doll. While its easy to deem the entire anime itself as a massive orgy between 7 doll sisters, nothing beats the rivalry between the 2 flagship dolls of the series, Shinku and Suigintou. Shinku, the 5th doll, was known as one of the smartest, calculative of the sisters, while Suigintou, the first doll, was made to be the most human like, possessing a sadistic and cruel personality. The 2 never liked each other from the start, and they constantly went at each other's throats.
Shinku tries to avoid as much conflict as possible within her sisters, Suigintou STRIVES to kill and bathe in the blood of her sisters to collect her Roza Mysticas. Shinku fights with rose petals and Suigintou with her deadly black angel feathers. Both are very different and see things VERY differently, but still, they ARE sisters, and you can tell. When they're not fighting together they usually spend time doing rather similar things, and throw the occasional smart ass remark at one another. At least for Suigintou's case, you can see that she still cares for her master (she looks after Megu all the time).
Adam Blade & Adam Arclight
Two brothers that were created to be the perfect Needless, Blade and Arclight were nothing more than experiments made by power hungry scientists. The Adam project, the Adams were experiments, nothing else. There were a ton of them created, but very few were actually successful. Other than Blade and Arclight, there was Saten. Other than that, there were very little else. They may not be blood related, but they were created by the same people under the same circumstances. As much they hate it, they're related.
Blade is one of the main protagonists of "Needless", while Arclight is the main antagonist of the show. How fitting is that? Anyway, Arclight is wanted dead by most of the needless around the world because of his tyrannical rule. Blade however, is just a random priest who INTENDS to someday take down Arclight. Both have the ability to memorize the abilities of other needless, which adds to some crazy shenanigans when they do battle. It all boils down to who memorized the better abilities and who has more abilities, and as expected, its an insane fight.