Motion mapping between characters with different structures but corresponding to homeomorphic graphs, meanwhile preserving motion semantics and perceiving shape geometries, poses significant challenges in skinned motion retargeting. We propose M-R2ET, a modular neural motion retargeting system to comprehensively address these challenges. The key insight driving M-R2ET is its capacity to learn residual motion modifications within a canonical skeleton space. Specifically, a cross-structure alignment module is designed to learn joint correspondences among diverse skeletons, enabling motion copy and forming a reliable initial motion for semantics and geometry perception. Besides, two residual modification modules, i.e., the skeleton-aware module and shape-aware module, preserving source motion semantics and perceiving target character geometries, effectively reduce interpenetration and contact-missing. Driven by our distance-based losses that explicitly model the semantics and geometry, these two modules learn residual motion modifications to the initial motion in a single inference without post-processing. To balance these two motion modifications, we further present a balancing gate to conduct linear interpolation between them. Extensive experiments on the public dataset Mixamo demonstrate that our M-R2ET achieves the state-of-the-art performance, enabling cross-structure motion retargeting, and providing a good balance among the preservation of motion semantics as well as the attenuation of interpenetration and contact-missing.