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It is demonstrated that exact diagonalization of the microscopic many-body Hamiltonian via systematic full configuration-interaction (FCI) calculations is able to predict the spectra as a function of detuning of three-electron hybrid qubits based on GaAs asymmetric double quantum dots. It is further shown that, as a result of strong inter-electron correlations, these spectroscopic patterns, including avoided crossings between states associated with different electron occupancies of the left and right wells, are inextricably related to the formation of Wigner molecules. These physical entities cannot be captured by the previously employed independent-particle or Hubbard-type theoretical modeling of the hybrid qubit. We report remarkable agreement with recent experimental results. Moreover, the present FCI methodology for multi-well quantum dots can be straightforwardly extended to treat Si/SiGe hybrid qubits, where the central role of Wigner molecules was recently experimentally confirmed as well.