Soft-bodied organisms such as octopuses and elephant trunks exhibit remarkable morphological adaptability, dynamically reconfiguring body shape and stiffness, and flexibly adjusting their control strategies to enable versatile behaviors. Inspired by these biological systems, various soft robots have emerged in recent decades, featuring diverse materials, stiffnesses, and morphologies tailored to specific tasks. Despite substantial advances in the materials and structural designs of soft robots, developing a generalizable control framework capable of rapid adaptation across diverse configurations remains a long-standing challenge. Existing controllers are limited to fixed configurations, demanding laborious configuration-specific remodelling and policy redesign for new configurations. Here, we introduce a generalizable control system that enables rapid adaptation across diverse soft robot configurations via reinforcement learning in a shared linear Koopman embedding space. By encoding robot dynamics into this embedding space, our method decouples control policies from specific morphologies, allowing real-time, model-free policy adaptation across diverse configurations without retraining from scratch. We validate our system across 33 distinct robot configurations. Our system achieves a 75 times reduction in transfer samples across configurations, while sustaining robust performance under high-speed motion, heavy payloads, and multiactuator faults, and achieving real-world skills previously unattainable in soft robotics. This work establishes a unified and adaptable control paradigm for diverse soft robot configurations, bridging mechanical reconfigurability with control flexibility, and may offer broader insights for generalizable control in complex physical systems.