Advances in multi-modal large language models (MLLMs) have inspired time series understanding and reasoning tasks. Recent attempts hybridize numerical time series with their visualized plots, facilitating precise value reasoning and visual structure comprehension for comprehensive time series understanding of MLLMs. However, effective numerical-visual modality integration remains challenging due to fine-grained temporal misalignment across modalities and severe entanglement between shared and modality-specific semantics, which hinder localized interpretation and complementary reasoning. To address these issues, we propose MADI, a multi-modal LLM enhanced with fine-grained alignment and disentangled interaction, featuring (1) Patch-level Alignment, which enforces physically grounded fine-grained correspondence across heterogeneous modalities, (2) Discrete Disentangled Interaction, which separates modality-common semantics into compact discrete latents and adaptively synergizes the purified modality-unique information, and (3) Critical-token Highlighting, which emphasizes informative, query-relevant signals for robust reasoning. Experiments on synthetic and real-world benchmarks show that MADI consistently outperforms general-purpose LLMs and time-series-specialized MLLMs. The code is available at https://github.com/usail-hkust/MADI.