The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $|0\rangle |\psi\rangle\mapsto|\psi\rangle |\psi\rangle$ for arbitrary $|\psi\rangle$.
However, in classical circuits we can easily perform the copy $(0, b)\mapsto(b, b)$ by the function $f:(a, b)\mapsto (a\oplus b, b)$.
Now obviously, the no-cloning theorem doesn't apply because these are bits and not qubits, but the real question concerns the following dilemma. The above classical circuit can be physically realized, and the physical realization has to be ultimately reducible to quantum mechanics. If quantum mechanics truly explains our world, it must be able to give an account of the behavior of bits (not just qubits), and therefore it must be able to give an account of how bits can be cloned.
So I suppose my real question is, what is the quantum mechanical account of how classical bits can be copied?
There is a related question asking how is it possible that classical computations are non-unitary. According to the answer, unitarity only applies to isolated quantum systems and any open system (such as a system in which measurements are done) does not necessarily obey unitarity (this is similar to how open systems do not necessarily obey conservation of energy).
This gives me a partial answer, but it doesn't give me a deep enough understanding. For example, I can understand how measurements lead to the apparent destruction of information, and so I can see how a classical gate can collapse bits (e.g. see XOR-gate $(a, b)\mapsto (a\oplus b)$ or AND-gate $(a, b)\mapsto (a\wedge b)$). But I don't see how this could give an account of cloning bits.
How can this dilemma be resolved in the context of cloning bits?