Identity recognition (e.g., person, animal re-identification) has traditionally relied heavily on static appearance cues. Yet motion--consistent, individual-specific dynamics--can provide a complementary and potentially more robust signature, especially when appearance is weak or variable. This raises a fundamental question: when identity-specific motion cues are clearly present, to what extent do modern video models use them for recognition? To investigate this question, we conduct a systematic diagnostic study and introduce BALLER120, a controlled benchmark of 120 professional basketball players performing free-throws. By focusing on the same multi-phase action across individuals, BALLER120 reduces action-level variation and identity-correlated acquisition biases, enabling fine-grained analysis of identity-specific kinematic patterns. We find that modern video models can predict identity accurately from RGB videos, but often rely on static appearance cues such as faces and jersey regions, even when informative motion cues are available. Strikingly, when appearance is suppressed through silhouette-only or skeleton-only inputs, the same model architectures shift toward motion micro-patterns (e.g., foot placement and elbow bending). Despite containing less visual information, appearance-suppressed representations achieve competitive accuracy and stronger robustness to appearance shifts. Our qualitative analyses further show that appearance-suppressed models attend to distinctive motion patterns across individuals. Overall, our study demonstrates that identity-specific motion signatures are present, informative, and learnable, but modern video models may overlook them in favor of easier static shortcuts unless appearance cues are explicitly suppressed.