Though the question was asked almost 2 years ago, yet since it hasn't been closed yet, maybe the following answer will be helpful.
Given your 2 equations
- $$U|\psi_1\rangle|\alpha\rangle = |\psi_1\rangle|\psi_1\rangle$$
- $$U|\psi_2\rangle|\alpha\rangle = |\psi_2\rangle|\psi_2\rangle$$
take the inner product of (1) with (2):
left-hand side: $$\langle \psi_1|\langle \alpha|U^\dagger U |\psi_2\rangle|\alpha\rangle = \langle \psi_1|\langle \alpha |\psi_2\rangle|\alpha\rangle = \langle \psi_1|\psi_2\rangle \langle \alpha |\alpha\rangle = \langle \psi_1|\psi_2\rangle $$
right-hand side: $$ \langle \psi_1| \langle \psi_1| \psi_2\rangle |\psi_2 \rangle = \langle \psi_1|\psi_2\rangle ^2 $$
Notice that the results on the left-hand side and on the right-hand side differ while they should be equal if cloning were possible. In fact, they are equal only in 2 cases:
If the states $|\psi_1\rangle$ and $|\psi_2\rangle$ are orthogonal, i.e. $\langle \psi_1|\psi_2\rangle = 0$
If the states $|\psi_1\rangle$ and $|\psi_2\rangle$ are identical, i.e. $\langle \psi_1|\psi_2\rangle = 1$
In other words, only orthogonal states can be cloned by the same unitary transformation.
You may also want to refer to "Introduction to Quantum Physics and Information Processing" by Radhika Vathsan where this proof is presented in slightly different form.