I firmly believe Vitani was born in the Pride Lands, and Nala would've seen Vitani when she was an infant cub. I can only speculate that Nala remembered Vitani because Vitani and the Outsider lionesses frequently intruded into the Pride Lands, and got in the Pridelanders's way when they were hunting. Since hunting prey is so hard to do, it would have irritated Nala a bit. It sounds silly even from me to have a grudge against someone over meat, but it's a fan theory, and no one's fan theories are correct. But, it's the best of as to why they both have a thing against each other.
I don't believe they both share the same father; it's possible, but I doubt it! And, don't get me started on that theory of Nala being Vitani's mother!
As for their genetic traits, they may share the same pelt fur, but Nala gets her's from Sarafina, and Vitani's from Zira. Sarafina and Zira, as far as I'm concerned, are unrelated. As for her eyes, they are actually supposed to be
green. They were that way when Nala was a cub, and I don't know why it didn't completely carry over when she matured into an adult, though a couple of shots of Nala as an adult have her with green eyes. I blame the lighting of the scene as a reason her eyes change so frequently. So, it is an obvious mistake on the TLKII artists's part to color her eyes light blue.