Interface problems are ubiquitous in science and engineering, arising in applications such as composite heat conduction, porous media flow, and biological transport, where heterogeneous subdomains interact through discontinuous coefficients and interface jump conditions. Accurately resolving such discontinuities is critical, as errors near interfaces often dominate the global solution. While classical mesh-based methods can achieve high accuracy, they face significant challenges for complex or irregular interface geometries due to mesh generation and refinement costs. Recent advances in physics-informed machine learning (PIML) offer mesh-free alternatives, but most existing approaches for interface problems are based on physics-informed neural networks (PINNs) and require retraining whenever problem parameters change, limiting their applicability to parametric settings. In this work, we introduce ϕ-DeepONet, a novel neural operator framework for learning solution operators of elliptic interface problems with discontinuous coefficients. Unlike prior domain-decomposition based operator learning methods, which suffer from rapidly increasing model complexity as the number of interfaces grows, the proposed approach embeds discontinuities directly into the operator architecture through a learned latent representation. Building on multi-input operator learning ideas, discontinuous input functions are handled using multiple branch networks, while interface information is encoded via a one-hot representation of subdomains that is projected into a low-dimensional latent space. This latent variable augments the output function space and is incorporated into a modified trunk network, enabling the operator to represent continuous solution manifolds without explicit domain decomposition or mesh dependence. The proposed ϕ-DeepONet framework is fully mesh-free, scalable to problems with multiple interfaces, and capable of generalizing across parametric variations without retraining. Numerical experiments on elliptic interface problems demonstrate that the method accurately captures solution discontinuities and interface behavior while maintaining computational efficiency. These results highlight the potential of ϕ-DeepONet as a robust operator-learning paradigm for a broad class of interface problems in scientific computing.