The key word there is continuous-spectrum. A free electron can have any energy, and therefore the energy eigenstates are not physically realizable, due to the n

The key word there is continuous-spectrum. A free electron can have any energy, and therefore the energy eigenstates are not physically realizable, due to the normalization requirements. But a bound electron, like in a quantum harmonic oscillator, has quantized energies and therefore can assume those eigenstates exactly. Think about electron orbitals in molecules: very precise energies.