Skeleton-based person re-identification (Re-ID) is an emerging open topic providing great value for safety-critical applications. Existing methods typically extract hand-crafted features or model skeleton dynamics from the trajectory of body joints, while they rarely explore valuable relation information contained in body structure or motion. To fully explore body relations, we construct graphs to model human skeletons from different levels, and for the first time propose a Multi-level Graph encoding approach with Structural-Collaborative Relation learning (MG-SCR) to encode discriminative graph features for person Re-ID. Specifically, considering that structurally-connected body components are highly correlated in a skeleton, we first propose a multi-head structural relation layer to learn different relations of neighbor body-component nodes in graphs, which helps aggregate key correlative features for effective node representations. Second, inspired by the fact that body-component collaboration in walking usually carries recognizable patterns, we propose a cross-level collaborative relation layer to infer collaboration between different level components, so as to capture more discriminative skeleton graph features. Finally, to enhance graph dynamics encoding, we propose a novel self-supervised sparse sequential prediction task for model pre-training, which facilitates encoding high-level graph semantics for person Re-ID. MG-SCR outperforms state-of-the-art skeleton-based methods, and it achieves superior performance to many multi-modal methods that utilize extra RGB or depth features. Our codes are available at https://github.com/Kali-Hac/MG-SCR.