Does an individual share of a stock have some kind of unique identifier?

Nope, think what a nightmare that would be, a bunch of shares would be issued and then sold to tonnes of people, who might sell various partial numbers of them to others, who might buy them and others from 20 others all as part of one order though multiple fills... It would be nuts, and if one were to issue a certificate with the IDs of shares that were carried through such a process the likelihood is the fragmentation would be so great that 100K shares would have consist of almost as many fragments! Imagine a share certificate with 70K IDs/ranges? Yikes!

There is no unique identifier that exists to identify specific shares of a stock. Just like money in the bank, there is no real reason to identify which exact dollar bills belong to me or you, so long as there is a record that I own X bills and I can access them when I want. (Of course, unlike banks, there is still a 1:1 relationship between the amount I should own and the amount they actually hold).

If I may reach a bit, the question that I assume you are asking is how are shared actually tracked, transferred, and recorded so that I know for certain that I traded you 20 Microsoft shares yesterday and they are now officially yours, given that it's all digital. While you can technically try and request a physical share certificate, it's very cumbersome to handle and transfer in that form.

Ownership of shares themselves are tracked for brokerage firms (in the case of retail trading, which I assume is the context of this question as we're discussion personal finance). Your broker has a record of how many shares of X, Y, and Z you own, when you bought each share and for how much, and while you are the beneficial owner of record (you get dividends, voting rights, etc.) your brokerage is the one who is "holding" the shares. When you buy or sell a stock and you are matched with a counterparty (the process of which is beyond the scope of this question) then a process of settlement comes into play. In the US, settlement takes 3 working days to process, and technically ownership does not transfer until the 3rd day after the trade is made, though things like margin accounts will allow you to effectively act as if you own the shares immediately after a buy/sell order is filled.

Settlement in the US is done by a sole source, the Depository Trust & Clearing Corporation (DTCC). This is where retail and institutional trade all go to be sorted, checked and confirmed, and ultimately returned to the safekeeping of their new owners' representatives (your brokerage). Interestingly, the DTCC is also the central custodian for shares both physical and virtual, and that is where the shares of stock ultimately reside.