Data-driven engineering could be remarkably empowered by the ability to disentangle uncertainties. However, simultaneously estimate epistemic uncertainty (model error) and aleatoric uncertainty (data noise) has remained extremely difficult. We propose a novel methodology explicitly designed to solve the uncertainty disentanglement conundrum: Variance estimation Bayesian Neural Networks (VeBNNs). The core innovation lies in a three-step cooperative learning strategy that integrates traditional training with probabilistic learning: (1) the base architecture is trained using the Mean Squared Error (MSE) loss (a standard practice); (2) the network is then trained using a novel Gamma loss specifically designed to learn the parameters of the aleatoric uncertainty (data noise); and (3) Bayesian inference is performed to achieve a robust prediction with uncertainty quantification and disentanglement. This approach is general, applicable to datasets ranging from simple single-fidelity models to complex history-dependent multi-fidelity scenarios. We demonstrate the versatility and effectiveness of VeBNNs across diverse applications, including several data-driven constitutive modeling problems. Most notably, we showcase the method's potential by successfully guiding the design of a sustainable polymer mixture of recycled polyethylene and polypropylene that optimizes toughness. We believe that VeBNNs provide a powerful and practical tool for robust analysis, material discovery, and decision-making under uncertainty.