Circuit quantum electrodynamics systems are typically built from resonators and two-level artificial atoms, but the use of multi-level artificial atoms instead can enable promising applications in quantum technology. Here we present an implementation of a Josephson junction circuit dedicated to operate as a V-shape artificial atom. Based on a concept of two internal degrees of freedom, the device consists of two transmon qubits coupled by an inductance. The Josephson nonlinearity introduces a strong diagonal coupling between the two degrees of freedom that finds applications in quantum non-demolition readout schemes, and in the realization of microwave cross-Kerr media based on superconducting circuits.