At the risk of the negative karma for entertaining this now exhausting topic…
I believe consciousness is the inflection upon the “substrate” of our minds. This theory suggests all matter is DORMANT consciousness animated by our electro-chemical jelly brains. Consciousness is literally the universe peering back upon itself, animated by living things.
So consciousness is more like electricity (potential of existential being) than a personal “item”, merely subjectively isolated to the local scope of our brains.
Thus there is no reason what you say couldn’t be true, that multiple atoms (animated through biotechnology) may be entangled and share “identity”. I think this is what is happening in our own minds to produce the coherent singularity of personal consciousness.
Further, every atom DOES have serial “numbers”, the extra-dimensional vibration suggested in string theory (whether or not that postulation is complete or wholly accurate). There would be more vibrational combinations than atoms in the universe. Synchronization of these vibrations would explain entanglement (two atoms in different locations sharing the same serial.)
Further further, null hypothesis proof requirements are absurd, you cannot predict or calculate all the ways a piece of paper may be folded let alone how reality may be folded to produce what wonders we perceive. This and that Truth is not confined to the horizon of our ignorance, provability and useful conjecture are not in parity. Otherwise no one would accept theories of … the Big Bang, dark matter/energy, infinite primes infinitely far apart, etc.