Ah, you've discovered separate but identical! The well-educated and neutral parties understand this.
If, in fact, we even need to go that far.
The property of self-awareness is one part of the discussion. It is one of myriad phenomena associated with an operating brain. Mine would not differ from that of a perfect copy of me. But here's a radical concept: mine doesn't differ from yours either. If we consider self-awareness as a property of an operating mind, and the essence of what separates a human from, say, a rock, then that mere property is not individualized. I'm self-aware. You're self-aware. Jabba is self-aware. But these aren't individualized or particular any more than "going 60 mph" is individualized for each car on the freeway, or any different among the different makes of car. A Volkswagen is going 60 mph and the Toyota in the next lane is also going 60 mph. The nature of "going 60 mph," as any other such property, isn't any different depending on where, when, or by whom it's manifest. There isn't any room in the concept of "going 60 mph" for individualization, nor need for any. Nor, I propose, is there in "is self-aware."
Now what we propose to duplicate by duplicating the organism is accumulated memory from the life up to that point. Being self-aware includes access to memories stored in the physical organism of the brain. And if Jabba's going to invoke reincarnation, then the sine qua non of that belief is the persistence of belief from one incarnation to the next. Incarnation aside, materialism has quite a ready explanation for how memories are stored in the brain and it's purely material. Sensory experiences, narratives, and emotional responses all have their easily-identified roots in the physicality of the brain. That is what would need to be hypothetically reproduced to arrive at an organism that is not only self-aware (because it has any old functioning brain) but would also, along with that self-awareness, also have access to all that the previous organism was up to the point of separation.
But note that the individualization is still in the organism, not in some trumped-up pseudo-philosophy. Jabba is trying to bundle up the portions of some particular life experience that are individualized (i.e., an individual's memories of life up to that point) and include them with this ineffable soul. In materialism all that
can be individualized is firmly seated in the organism, not "me" or "the self" or any of the other lofty language he's borrowing from real philosophers.