We propose a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for input and output function distributions. Our method differs from existing methods like DeepONet and POD-NN/PCA-Net in its applicability to arbitrary input and output discretizations with a single model that is trained end-to-end. Our model is discretization-independent, making it particularly effective for multifidelity learning. We establish theoretical approximation guarantees, demonstrating uniform universal approximation under strong assumptions on the input functions and statistical approximation under weaker conditions. We validate our method through extensive numerical experiments involving both local and nonlocal PDEs, including time-independent and time-dependent problems. The results show that multifidelity training significantly improves accuracy and computational efficiency. Moreover, multifidelity training further enhances empirical discretization independence.