Medical brain imaging relies heavily on image registration to accurately curate structural boundaries of brain features for various healthcare applications. Deep learning models have shown remarkable performance in image registration in recent years. Still, they often struggle to handle the diversity of 3D brain volumes, challenged by their structural and contrastive variations and their imaging domains. In this work, we present NeuReg, a Neuro-inspired 3D image registration architecture with the feature of domain invariance. NeuReg generates domain-agnostic representations of imaging features and incorporates a shifting window-based Swin Transformer block as the encoder. This enables our model to capture the variations across brain imaging modalities and species. We demonstrate a new benchmark in multi-domain publicly available datasets comprising human and mouse 3D brain volumes. Extensive experiments reveal that our model (NeuReg) outperforms the existing baseline deep learning-based image registration models and provides a high-performance boost on cross-domain datasets, where models are trained on 'source-only' domain and tested on completely 'unseen' target domains. Our work establishes a new state-of-the-art for domain-agnostic 3D brain image registration, underpinned by Neuro-inspired Transformer-based architecture.