Soft robots offer remarkable adaptability and safety advantages over rigid robots, but modeling their complex, nonlinear dynamics remains challenging. Strain-based models have recently emerged as a promising candidate to describe such systems, however, they tend to be high-dimensional and time-consuming. This paper presents a novel model order reduction approach for soft and hybrid robots by combining strain-based modeling with Proper Orthogonal Decomposition (POD). The method identifies optimal coupled strain basis functions -- or mechanical synergies -- from simulation data, enabling the description of soft robot configurations with a minimal number of generalized coordinates. The reduced order model (ROM) achieves substantial dimensionality reduction in the configuration space while preserving accuracy. Rigorous testing demonstrates the interpolation and extrapolation capabilities of the ROM for soft manipulators under static and dynamic conditions. The approach is further validated on a snake-like hyper-redundant rigid manipulator and a closed-chain system with soft and rigid components, illustrating its broad applicability. Moreover, the approach is leveraged for shape estimation of a real six-actuator soft manipulator using only two position markers, showcasing its practical utility. Finally, the ROM's dynamic and static behavior is validated experimentally against a parallel hybrid soft-rigid system, highlighting its effectiveness in representing the High-Order Model (HOM) and the real system. This POD-based ROM offers significant computational speed-ups, paving the way for real-time simulation and control of complex soft and hybrid robots.