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Does an individual share of a stock have some kind of unique identifier?
I agree with the answer by @Michael that this number doesn't exist.  It's hard to see what use it would have and it would be difficult to track. I'm writing a separate answer because I also disagree with the premise of your question:  Individual shares of stock have never to my knowledge had such a number.  Your comment about numbers on stock certificates identifies the certificate document, which will generally represent multiple shares of stock.  That number no more identifies a single share of stock than the serial number on a $10 bill identifies any one of the ten dollars it represents.  Even at the "collective" unit of $10, when the bill is eventually replaced with a new one, the new bill has a new number.  No continuity.