Learning a robust Variational Autoencoder (VAE) is a fundamental step for many deep learning applications in medical image analysis, such as MRI synthesizes. Existing brain VAEs predominantly focus on single-modality data (i.e., T1-weighted MRI), overlooking the complementary diagnostic value of other modalities like T2-weighted MRIs. Here, we propose a modality-aware and anatomically grounded 3D vector-quantized VAE (VQ-VAE) for reconstructing multi-modal brain MRIs. Called NeuroQuant, it first learns a shared latent representation across modalities using factorized multi-axis attention, which can capture relationships between distant brain regions. It then employs a dual-stream 3D encoder that explicitly separates the encoding of modality-invariant anatomical structures from modality-dependent appearance. Next, the anatomical encoding is discretized using a shared codebook and combined with modality-specific appearance features via Feature-wise Linear Modulation (FiLM) during the decoding phase. This entire approach is trained using a joint 2D/3D strategy in order to account for the slice-based acquisition of 3D MRI data. Extensive experiments on two multi-modal brain MRI datasets demonstrate that NeuroQuant achieves superior reconstruction fidelity compared to existing VAEs, enabling a scalable foundation for downstream generative modeling and cross-modal brain image analysis.