He wanted to be more than half of the trademarked BeckandJade, JadeandBeck. When girls smiled at him, he didn't want to have to hear the whispers of their friends warning, "That's Beck. Jade's Beck!" He wanted to be more than the boy who tamed the wild beast (or didn't, because Jade was not and never would be tame).
He did love Jade, and he loved being with her despite how taxing their relationship sometimes was. But he hated feeling like he was another of her playthings, another pawn on the chessboard that she used to manipulate the entire world around her. She was careful not to push him, not to force him or coerce him, but she did manipulate him. She probably didn't even realize she was doing it, but Jade was Jade because she ruled with an iron fist, the common people doing as she directed below her.
He didn't want to be the king to her queen of broken hearts. She had a legacy of her own and he respected that, but he wanted to have his own legacy.
He wanted those things, but he never tried too hard to obtain them. When someone would ask "BeckandJade" to go somewhere or say "Hi, BeckandJade," the little lines of annoyance on her face would smooth out into flawless porcelain skin and the corner of her lips would ever so subtly inch upwards in satisfaction. And he wondered if it would even be worth being Beck, no matter how his web of talent and empathy and handsomeness spread around the world, without Jade there to hold his hand and smirk at him because she knew that he knew and they knew everyone knew that he was hers.