The Replay file posted by Dack answers it all for me.
Look at all the lines in the file that say "<city> destroyed".
The next line in the file is almost invariably "<same Civ> found the city of ...".
I postulate the following:
(1) The AI has a new settler because he just destroyed a city producing it.
(2) At this point the "hey this looks like a good spot for a city" logic kicks in and the settler builds a city in the same spot.
(3) The cheap settlers and ready cash make building settlers favorable in the new city so the pattern repeats.
The Americans are the only ones not destroying cities, so I assume this is kirkham7.
I was always under the impression that once you got to a point where you used all the names that it started using unused city names from other civs. IE i was german and they offered me the city name of Mecca.
The first civs to run through their 16-city lists were the Zulus and the Indians.
The Zulus start on the Mecca list in 3600 followed closely by the Indians in 3580.
By the time the Zulus found Sydney (end of the Mecca list) in 3360, everyone but the Greeks (16th in 3320) and Russians (16th in 3240) have built their 16 cities.
With all Mecca-Sydney cities founded, the game has to go looking elsewhere for city names.
The first available is the Roman list.
In 3360, the French found Rome. This city is destroyed in 3300. Since there is no record of its capture and AIs don't rename cities, we can assume the rename to Yolo happened later.
The Roman cities are used up in 3280; Babylonian in 3220; German in 3140.
The next available city list in the Aztec cities.
With all 7 civs building cities at a frenetic pace, the last 64 city names are used up by 2880.
With all 256 city names used up, the game falls back on its "array overflow" logic:
Use the first element in the city name array (initially "Rome").
This is confirmed by the fact that renaming any of the post-Kerman "Rome"s to "Yolo" renames them all.
All-in-all, this is the most informative test of the game logic I've seen in years.