What is quantum about the no-cloning theorem?

The answer seems to be given here.

Let me summarize it: the point is that the notion of copying that is usually seen (such as in the OP) is actually not what we mean by copying. It only allows for the object that we want to copy, and the new object. In that case it is indeed impossible to clone. But the linked article shows that if we also include a copier, then it becomes possible to clone in the classical case (but remains impossible in the quantum case).

(Note: he actually doesn't prove that it's always possible classically, but at least he gives some explicit examples where it is.)

You're right that no-cloning isn't inherently quantum. The classical analogue of quantum cloning is "probability cloning".

Here's the probability cloning challenge: create a machine that takes one coin prepared to be heads up with probability $p$, but outputs two coins that are heads up with probability $p$. The catch is that you don't know $p$ (the machine should work for all $p$) and the two output coins are supposed to be independent. The statistics of the output should satisfy $P(HH) = p^2$, $P(HT) = P(TH) = p (1-p)$, and $P(TT) = (1-p)^2$.

I hope you see immediately why this machine can't possibly exist (or, at least, must perform very poorly). The coin's state doesn't tell you enough about the probability that generated it. One sample just isn't enough information to get a good idea of the underlying distribution. So the best you can do is make the two coins match the input, maybe throw in a bit of noise, and hope the statistical tests miss the obvious.

Another way to think about it is that there's no stochastic matrix $S$ that satisfies:

$$S \begin{bmatrix} p \\ 1-p \end{bmatrix} = \begin{bmatrix} p \\ 1-p \end{bmatrix}^{\otimes 2} =\begin{bmatrix} p^2 \\ p (1-p) \\ p(1-p) \\ (1-p)^2 \end{bmatrix}$$

This is obvious because the output requires components of the input to be multiplied and squared. You can't go from $p$ to $p^2$ using a linear operation.

See also: the 2002 paper 'Classical No-Cloning Theorem' by A. Daffertshofer et al., which is cited by the paper you cited.

Comments to the post (v1):

1. So there are at least 4 related theorems:

   1. The quantum No-cloning theorem for pure states by Wootters, Zurek & Dieks.

   2. The quantum No-broadcast theorem for density operators.

   3. The classical symplectic No-cloning theorem by Refs. 1-2. This is what OP asks about.

   4. The classical probabilistic No-cloning theorem using Kullback-Leibler divergence by Ref. 3. We will have nothing more to say about item 4.

2. Note that both Thm. 1 & Thm. 3 have elementary one-line proofs. It might therefore not be very constructive/productive/useful/meaningful to try to compare them. [E.g. such a 'detail' of whether we consider (i) a Hilbert space or (ii) a projective Hilbert space/ray space (as we do in quantum mechanics) is already more subtle, cf. e.g. Wigner's theorem & its non-trivial proof.]

3. In geometric quantization, under the correspondence principle between classical & quantum mechanics, the Hilbert space ${\cal H}$ is identified with a Lagrangian submanifold via polarization. Therefore we cannot use the Hilbert space ${\cal H}$ to probe non-zero directions of the symplectic structure. Hence we may argue that the cloning obstruction in Thm. 3 does not reflect the cloning obstruction in Thm. 1.

4. Rather Thm. 1 is of a different nature, possibly closer related to Thm. 2. If time permits, we may elaborate further on this point in the future.

References:

1. A. Fenyes, There’s no cloning in symplectic mechanics, 2010.

2. A. Fenyes, J. Math. Phys. 53 (2012) 012902, http://arxiv.org/abs/1010.6103.

3. A. Daffertshofer, A. R. Plastino, & A. Plastino, Phys. Rev. Lett. 88 (2002) 210601.