I'm reading Wadler's 'Theorems for free'. In section 3.5 he states that $m_{AA}(I_A)$ is a rearranging (i. e. injective) function. $I_A$ is the identity function on the type A. $$m : \forall X.\forall Y. (X \rightarrow Y) \rightarrow (X^* \rightarrow Y^*)$$ How does he use section 3.4 to prove this without any $\forall^{(=)}X$-types? Why can't $m_{AA}(I_A)$ e. g. drop a few elements in the list?

It can, as good evidence for it define m as something like
m f = map f . drop 2

m (\x -> x) . map f = map f . m (\x -> x)

m (\x -> x) performs all the structural changes to the list and map f actually alters this element. This theorem implies that they are independent of each other.