Predicting future trajectories of pairwise traffic agents in highly interactive scenarios, such as cut-in, yielding, and merging, is challenging for autonomous driving. The existing works either treat such a problem as a marginal prediction task or perform single-axis factorized joint prediction, where the former strategy produces individual predictions without considering future interaction, while the latter strategy conducts conditional trajectory-oriented prediction via agentwise interaction or achieves conditional rollout-oriented prediction via timewise interaction. In this article, we propose a novel double-axis factorized joint prediction pipeline, namely, conditional goal-oriented trajectory prediction (CGTP) framework, which models future interaction both along the agent and time axes to achieve goal and trajectory interactive prediction. First, a goals-of-interest network (GoINet) is designed to extract fine-grained features of goal candidates via hierarchical vectorized representation. Furthermore, we propose a conditional goal prediction network (CGPNet) to produce multimodal goal pairs in an agentwise conditional manner, along with a newly designed goal interactive loss to better learn the joint distribution of the intermediate interpretable modes. Explicitly guided by the goal-pair predictions, we propose a goal-oriented trajectory rollout network (GTRNet) to predict scene-compliant trajectory pairs via timewise interactive rollouts. Extensive experimental results confirm that the proposed CGTP outperforms the state-of-the-art (SOTA) prediction models on the Waymo open motion dataset (WOMD), Argoverse motion forecasting dataset, and In-house cut-in dataset. Code is available at https://github.com/LiDinga/CGTP/.