This largely depends on your viewpoint, the map M : A * A \to A was meant to be a morphism in an arbitrary monoidal category. An element like x, would be interpreted as a morphism x : I -> A (in general the objects A in a monoidal category, or any category for that matter have no notion of elements). The expression f(x) is interpreted as the composition f . x, so no you are not free to use x a second time, once you formed that composition. This is swept under the rug in most cases because monoidal categories like Set are cartesian and allow you to freely copy morphisms like x : I -> A, via the Delta map Delta : A -> A * A. Explicitly you have Delta(x) = Delta . x = x * x. In general it might not be possible to even determine if two morphisms x : I -> A and y : I -> A are the same.

In some cases you might be able to assume from the start that you have a certain number of equal morphisms x : I -> A "in reserve", for example in linear logic you have the operator ! ("of course") which gives you an arbitrary number of identical morphisms to play with.

In any case this distinction is quite subtle and I understand, why you might assume that I'm simply misunderstanding things. In particular I should emphasize that almost no programming languages work this way, although with some effort you would be able to recast typical cryptographic / numerical code in this language.

It is also really easy that to see for example in the case of addition that indeed information is destroyed, clearly the map

(a,b) -> (a+b,a-b)

has an inverse if the characteristic isn't 2, the subsequent projection to the first factor destroys information. Categories in which that is possible have maps d_A : A -> I.

What tel is pointing out above, is that Delta and d together form a comonoid.

While psi \otimes psi certainly makes sense, the map psi \to psi \otimes psi is not linear and therefore not a morphism (Physicists call this the no cloning theorem for some reason).