Optical and Synthetic Aperture Radar (SAR) image registration is crucial for multi-modal image fusion and applications. However, several challenges limit the performance of existing deep learning-based methods in cross-modal image registration: (i) significant nonlinear radiometric variations between optical and SAR images affect the shared feature learning and matching; (ii) limited textures in images hinder discriminative feature extraction; (iii) the local receptive field of Convolutional Neural Networks (CNNs) restricts the learning of contextual information, while the Transformer can capture long-range global features but with high computational complexity. To address these issues, this paper proposes a multi-expert learning framework with the State Space Model (ME-SSM) for optical and SAR image registration. Firstly, to improve the registration performance with limited textures, ME-SSM constructs a multi-expert learning framework to capture shared features from multi-modal images. Specifically, it extracts features from various transformations of the input image and employs a learnable soft router to dynamically fuse these features, thereby enriching feature representations and improving registration performance. Secondly, ME-SSM introduces a state space model, Mamba, for feature extraction, which employs a multi-directional cross-scanning strategy to efficiently capture global contextual relationships with linear complexity. ME-SSM can expand the receptive field, enhance image registration accuracy, and avoid incurring high computational costs. Additionally, ME-SSM uses a multi-level feature aggregation (MFA) module to enhance the multi-scale feature fusion and interaction. Extensive experiments have demonstrated the effectiveness and advantages of our proposed ME-SSM on optical and SAR image registration.