How is it possible to maintain classical data encoded into qubits, which often contains copies of information, given that the no cloning theorem prevents cloning information?
The no cloning theorem only applies when quantum information is in an unknown superposition. If you know a basis in which the state of some qubits is not under superposition, then you can make all the copies you want.
Classical information encoded directly into qubits is going to be in the computational basis state. Therefore you can clone it. You use CNOT operations to do it.