I understand where you are come from, but you are conflating different meta-levels (external vs. internal language/logic): You are absolutely free to use any formal expression "x" a second time, and - crucially - it will have the same mathematical meaning if you choose to do so, as can be seen by pondering over the tautology "P(x) <-> P(x)". This is not a property of the category that you chose to work with, but rather of the formal language that we use when engaging in mathematics, as you have demonstrated yourself when you had permitted that "psi \otimes \psi" made sense.
The non-existence of a morphism "psi -> psi \otimes psi" and the notion of "destroyed information" that you are discussing in the rest of your post is independent from all of this. If you wish I can elaborate on the "no cloning theorem".
Well ok, then this was a simple misunderstanding, or miscommunication. My knowledge is largely based on reading of http://ncatlab.org/nlab/show/internal+logic and several other pages on that wiki, plus graduate training in math and physics. Based on your last reply I'm confident you have a similar background. Logic and category theory are not my specialty and I probably haven't expressed myself as clearly as I would in professional communication.
The non-existence of a morphism "psi -> psi \otimes psi" and the notion of "destroyed information" that you are discussing in the rest of your post is independent from all of this. If you wish I can elaborate on the "no cloning theorem".