Dynamic and continuous jumping remains an open yet challenging problem in bipedal robot control. Real-time planning with full body dynamics over the entire jumping trajectory presents unsolved challenges in computation burden. In this paper, we propose a novel variable-model optimization approach, a unified framework of variable-model trajectory optimization (TO) and variable-frequency Model Predictive Control (MPC), to effectively realize continuous and robust jumping planning and control on HECTOR bipedal robot in real-time. The proposed TO fuses variable-fidelity dynamics modeling of bipedal jumping motion in different jumping phases to balance trajectory accuracy and real-time computation efficiency. In addition, conventional fixed-frequency control approaches suffer from unsynchronized sampling frequencies, leading to mismatched modeling resolutions. We address this by aligning the MPC sampling frequency with the variable-model TO trajectory resolutions across different phases. In hardware experiments, we have demonstrated robust and dynamic jumps covering a distance of up to 40 cm (57% of robot height). To verify the repeatability of this experiment, we run 53 jumping experiments and achieve 90% success rate. In continuous jumps, we demonstrate continuous bipedal jumping with terrain height perturbations (up to 5 cm) and discontinuities (up to 20 cm gap).