As I suspected it's about per capita GDP measured (as is today's curious fashion, or at least it was until recent shifts to a 2005 base) in "1990 international Geary-Khamis dollars" which neither Italians nor anyone else used in 1300 or even in 1870. The quote continues:

Historians estimate that the average annual income in Italy around the year 1300 was roughly $1,600. Some 600 years later – after Columbus, Galileo, Newton, the scientific revolution, the Reformation and the Enlightenment, the invention of gunpowder, printing, and the steam engine – it was... still $1,600.

I hate these calculations. Though credited here to Bolt & van Zanden, the figures here seem to be a variation of Paolo Malanima's estimates for northern and central Italy. As here it only concerns one area, the faddish anachronistic pricing is the least of it: here the problem is the computation's basis in deflated wages which the leading British economic historian Michael Postan said "weren't worth the paper they were printed on".

Malanima at least acknowledges that not everyone lived on wages - fewer still in the earlier period - but reports that an earlier study of peasant contracts suggests a similar trend (though its author warns that the relationships assumed in his results may not apply in the earlier period when customary arrangements were stronger). And he does a better job of it than most, giving us annual current-price estimates along with figures in more useful 1911 and (implicitly) 1861 lire along with his crucial price index, though this may overstate real income volatility as it appears to be based on consumption early in the period.

So I'm evidently not an aficionado of the method, which to me divides one possibly unrepresentative number by another. But what of the results? I've little faith in the movements between individual centuries for reasons indicated above, but the long-run trend probably offers a reasonable approximation to reality. That's because northern & central Italy in 1300 constituted one of Europe's most prosperous regions, half a millennium later one of the west's perceived laggards as the continent's economic centre of gravity shifted from the Mediterranean to the north (a process already under way at the start of the period) and the Atlantic. So in measuring income growth over the long run it's one of the most "pessimistic" regions you can cite.

Did Italian incomes really stagnate or even fall slightly? It's possible, and they certainly didn't rise dramatically as in northern economies. But I'm sceptical that it was quite so bad, not least because I find a modest rise in Italy's urbanisation level over the period (most of it admittedly after 1800) where Malanima sees a fall (Italian urban population is a headache because earlier figures tend to be for the commune which often extended far beyond the town proper), and urbanisation is an indicator of agrarian surplus. On the other hand, while maize added to the region's crops (though often at the expense of better grains), the rise in associated vitamin deficiency in the northeast and Lombardy indicates the persistence of severe poverty even as landowners profited - though this should remind us that labour and other incomes didn't necessarily move in the same direction.

So to answer your question, per capita income growth over the period seems likely to have been at best modest, and a slight decline can't be ruled out especially among the (still mostly agricultural) working population and especially to the early 19th century, though I’m far from convinced, particularly when we allow for possible growth of non-labour earnings. What happened after 1870? Malanima finds growth of about two-thirds in per capita income to 1913, Fenoaltea barely half, so you're right that an older early 20th-century observer would have seen more change in real earnings in under half a century than over the previous half-millennium.