The operation $𝑈|1⟩_𝐴|0⟩_𝐵=|0⟩_𝐴|1⟩_𝐵$$𝑈|\psi⟩_𝐴|\phi⟩_𝐵=|\phi⟩_𝐴|\psi⟩_𝐵$ (swapping the state of the two qubits) is a unitary operation. So, it is a valid quantum operation. It is known in quantum computing as the $\text{SWAP}$-gate and is represented by the matrix $$\text{SWAP} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
On the other hand, cloning means to create an independent and identical copy of an arbitrary unknown quantum state. That is:
$$𝑈|\psi⟩_𝐴|0⟩_𝐵=|\psi⟩_𝐴|\psi⟩_𝐵$$
No-cloning theorem states that such unitary operator $U$ does not exist.