Correct, cloning in even the simplest sense of a single particle state (let alone a whole human) is impossible in quantum mechanics.

Though cloning a human may be possible if the brain does not rely on quantum state. Obligatory "warm and wet" objection.

It sounds like state send is partial using the "phone" analogy.

Can there be triplicates of entanglement?

There are limits to how much three systems can be entangled with each other. It turns out that one system cannot be maximally entangled with two different systems. In fact, these types of arguments are often used in studying information paradoxes in black holes.

Yup https://en.wikipedia.org/wiki/W_state

Indeed there is a “no cloning theorem”.

Iirc it essentially comes from the fact that time evolution is linear, and a function that sends stateOfIntetest \otimes constantState to stateOfInterest \otimes stateOfInterest for all values of stateOfInterest would be quadratic?

Yes, it's a direct result from the linearity of time evolution in QM. Like you say, if state1 x raw_material -> state1 x state1 (i.e. we are able to take some raw material and clone state1) and also state2 x raw_material -> state2 x state2, then linearity forces us to also have (state1 + state2) x raw_material -> state1 x state1 + state2 x state2. This is not the same as (state1 + state2) x (state1 + state2) which is what we'd want in order to clone arbitrary states.

Hey yeah I remember reading Penrose's "The Emperor's New Mind" in maybe 1998?, an interesting take on the nature of consciousness... fascinating stuff.

I don't think you would need exact duplication for these situations to arise.