That's indeed an assumption. But so far there isn't much evidence pointing at the quantum solution, and there are theories of how self-awareness can arise that don't require magic. It's reasonable to take the less magical assumptions until presented with (appropriate amount of) evidence to the contrary.

Quantum-based self-awareness is a cop-out IMO, an open way to claim that it comes from God. Well, if that's true and it's beyond our reach, then we'll never build a self-aware AI ourselves. But assuming something is impossible doesn't help moving forward...