Accurate cardiac motion estimation from cine cardiac magnetic resonance (CMR) images is vital for assessing cardiac function and detecting its abnormalities. Existing methods often struggle to accurately capture heart motion because they rely on intensity-based image registration similarity losses that may overlook cardiac anatomical regions. To address this, we propose CardioMorphNet, a recurrent Bayesian deep learning framework for 3D cardiac shape-guided deformable registration using short-axis (SAX) CMR images. It employs a recurrent variational autoencoder to model spatio-temporal dependencies across the cardiac cycle, along with two posterior models for bi-ventricular segmentation and motion estimation. The derived loss function from the Bayesian formulation guides the framework to focus on anatomical regions by recursively registering segmentation maps without using intensity-based image registration similarity loss, while leveraging sequential SAX volumes and spatio-temporal features. The Bayesian modelling also enables the computation of uncertainty maps for the estimated motion fields. Validated on the UK Biobank and M&M datasets by comparing warped mask shapes with ground-truth masks, CardioMorphNet demonstrates superior performance in cardiac motion estimation, outperforming state-of-the-art methods. Uncertainty assessment shows that it also yields lower uncertainty values for estimated motion fields in the cardiac region compared with other probabilistic-based cardiac registration methods, indicating higher confidence in its predictions. In addition, the clinical indices extraction assessment shows that CardioMorphNet estimates the clinical indices more accurately than other approaches.