The quote referenced age of “great civilizations,” not of democracies.
However, the USA is the world’s oldest democracy, at 235 years or so. And there is a reasonable argument it didn’t become a true democracy till the 1960s.
Rome and Athens were never anything we would consider even a limited democracy. To get to 300 years for Roman “democracy,” even stretching various points, you have to go back to well before it was a “great civilization.” For most of the time is was “democratic” it was just one of many jostling city-states on the Italian peninsula.
There are no other democracies in history that have anything like 200 years under their belt. Republics, yes, but not democracies.
A clarification. Rome became something resembling a democracy, if you squint and look at it sideways, around 290, and the democracy ended at the latest, IMO, in 83 when Sulla conquered the city.
After that the Romans still held elections, but the legions controlled actual power and made the final decisions.
So you can make a reasonable case for Roman democracy lasting about 200 years, even as a “great civilization.”
Of course, you have to recognize that for this entire period the “democracy” was a rule by a master race of Roman citizens over a vastly outnumbering mass of slaves and subjects. Giving it a good bit of similarity to the National Socialist ideal.