This is interesting. I take this as another step towards New York City being recognized as a nation of it's own, much like The City [1] and Vatican City [2]. The United Nations HQ is already based there, so it has a good claim to said sovereignty.
I personally like the idea of independent city-states but historically they have not fared terribly well. I think you'll see a resurgence of this idea in outer space in the coming decades.
Vatican City on the other hand is a sovereign entity under the Lateran Treaty, and is recognized as such by international law.
Under US law it is very difficult to see how either would be possible without a Constitutional amendment.
By ancient custom the Lord Mayor tenders the City Sword to the Sovereign when she enters the City on state occasions, in token acknowledgement of her overriding authority. The Sovereign touches the sword and, by that gesture, returns it to the Lord Mayor with implied permission to carry it before her whilst she is in the City. Also by ancient custom the permission of the Lord Mayor is sought for the passage of troops through the City, and he receives quarterly, under the Sovereign's sign manual, the password of the Tower of London. As spokesman for the citizens, he is entitled to the right of special access to the Sovereign.
They're essentially a small enclave of "free" [1] people who are allied with Liz, smack dab in the middle of London. Most of the previous Lord Mayors of the City of London were descendants of the people who forced King John to sign the Magna Carta. If, in order to stay in honor of the rules of law, they must grant her a faux sovereignty in their kingdom, it's to do her honor that she would not normally [2] need bestowed on her, in order to keep her allied with The City.
[1] https://en.wikipedia.org/wiki/Freedom_of_the_City#Freedom_of...
[2] (Free people are all "kings" or "sovereign")