quantum Rabi model describes the simplest interaction of a two-level system with a single-mode boson field. When the coupling is arbitrary large, the bare multilevel structure of a realistic circuit atom cannot be ignored even if the circuit is strongly anharmonic. We explored this situation theoretically for flux (fluxonium) and charge (Cooper pair box) type multi-level circuit atoms at maximal frustration and identified which spectral features of the quantum Rabi model survive and which are renormalized for arbitrary large coupling. We provide a quantitative comparison with the ideal quantum Rabi model by inspecting not only the circuit energy level spectrum, but also the entanglement spectrum. Despite significant renormalization of the low-energy energy spectrum in the fluxonium case, the key quantum Rabi feature — nearly-degenerate vacuum consisting of an atomic state entangled with a multi-photon field — appears in both circuits when the coupling is sufficiently large. Like in the quantum Rabi model, for very large couplings the entanglement spectrum is dominated by only two, nearly equal eigenvalues, in spite of the fact that a large number of bare atomic states are actually involved in the ground state. We interpret the emergence of the vacuum degeneracy in both circuits as an environmental suppression of flux/charge tunneling due to their dressing by virtual low-/high-impedance photons in the resonator. For flux tunneling, the dressing is nothing else than the shunting of a Josephson atom with a large capacitance of the resonator. Suppression of charge tunneling appears to have the same origin as the dynamical Coulomb blockade of transport in tunnel junctions connected to resistive leads.