Abstract: Point defects are of interest for many quantum device applications such as spin qubits, single-photon emitters, and nanoscale sensors. Because of their localized orbitals, the electronic states are often strongly correlated, which has led to a proliferation of quantum embedding theories to treat this correlation. In embedding theories, a weakly correlated reference such as density functional theory is used to treat the full system, while certain one-particle states are singled out as an active space to be treated with an effective interaction. We assess embedding theories in the context of transition metal defects in aluminum nitride by referencing to fully correlated quantum Monte Carlo descriptions. These comparisons allow us to have access to detailed information about the many-body wave functions, which are not available experimentally. We find that errors in the underlying density functional theory treatment of correlations propagate to the embedded model's active space orbitals, effective interactions, and many-body excited states. These errors are extremely difficult to recover from by adding corrections such as double counting or many-body perturbation theory.