Aya is Tamaki's younger sister, and the two grew up together in an orphanage. When their family fell into serious debt, Kujou Takamasa adopted her—partly out of opportunity, partly out of ambition. He saw potential in her as an idol who could eventually surpass Zero, the same dream he'd instilled in Tenn. This adoption made Aya Tenn's sister as well, pulling her into Kujou's orbit and his carefully constructed plans.
For Tamaki, becoming an idol was always about one thing: finding Aya again after they were separated. It's his core motivation, the reason he pushes so hard. Meanwhile, Kujou has been training Aya abroad, mainly in America, developing her talent away from the usual spotlight.
Back when Kujou was working with Haruka, the two grew close—close enough that according to Kujou, they were girlfriends. That connection is part of Aya's history, though it's largely in the past now.
Content compiled by AnimeList.moe from publicly available sources.