Time series classification (TSC) of biological signals has progressed from handcrafted, modality-specific approaches to deep architectures capable of representing the diverse waveform structures of underlying physiological processes (i.e., morphology). This review introduces a unified morphology--modality framework that connects waveform structure to a methodological design, revealing how spikes, bursts, oscillations, slow drift, and hierarchical rhythms inform model design. By analyzing electroencephalography, electromyography, electrocardiography, photoplethysmography, and ocular modalities (electrooculography, pupillometry, eye-tracking), the review demonstrates how morphology determines preprocessing and modeling strategies. Integrating evidence across these biological signals, the framework reveals that morphology, not model class, most strongly determines performance and interpretability. This provides insight into why deep models succeed when their inductive biases align with underlying waveform dynamics. This review also identifies future work including morphological data augmentation and evaluation metrics to improve generalization. Together, these insights position morphology-aware modeling as a unifying principle for developing generalizable, interpretable, and physiologically meaningful TSC models across biological signals.