Graph-based models have emerged as a powerful paradigm for modeling multimodal urban data and learning region representations for various downstream tasks. However, existing approaches face two major limitations. (1) They typically employ identical graph neural network architectures across all modalities, failing to capture modality-specific structures and characteristics. (2) During the fusion stage, they often neglect spatial heterogeneity by assuming that the aggregation weights of different modalities remain invariant across regions, resulting in suboptimal representations. To address these issues, we propose MTGRR, a modality-tailored graph modeling framework for urban region representation, built upon a multimodal dataset comprising point of interest (POI), taxi mobility, land use, road element, remote sensing, and street view images. (1) MTGRR categorizes modalities into two groups based on spatial density and data characteristics: aggregated-level and point-level modalities. For aggregated-level modalities, MTGRR employs a mixture-of-experts (MoE) graph architecture, where each modality is processed by a dedicated expert GNN to capture distinct modality-specific characteristics. For the point-level modality, a dual-level GNN is constructed to extract fine-grained visual semantic features. (2) To obtain effective region representations under spatial heterogeneity, a spatially-aware multimodal fusion mechanism is designed to dynamically infer region-specific modality fusion weights. Building on this graph modeling framework, MTGRR further employs a joint contrastive learning strategy that integrates region aggregated-level, point-level, and fusion-level objectives to optimize region representations. Experiments on two real-world datasets across six modalities and three tasks demonstrate that MTGRR consistently outperforms state-of-the-art baselines, validating its effectiveness.