Italians living in the medieval and Renaissance periods were well aware of their identity as ‘Italians’ of ancient Roman heritage, e.g., language, culture, etc., but in answering your question, one has to consider the importance of campanilismo, or ‘hometown patriotism.’

After the fall of the Western Roman Empire, the Italian peninsula was not a unified political body—and won’t be until the Italian unification in the nineteenth century, i.e., the Risorgimento. In the medieval and Renaissance periods, an Italian would know themself to be Italian/Latin —Dante uses the word ‘latino’ to mean ‘italiano’ in the Purgatorio, for example —but would ultimately and more importantly consider themself as a Florentine, Milanese, Arentine, Sienese, Venetian, etc. In the same vein the ancient Greeks recognized their Hellenic culture and identity, there was no national identity: you were Greek, yes, but you were most importantly an Athenian; you were Italian, but a Perugian first, etc.

As I briefly mentioned earlier, this started to change with the formation of the ‘new’ Italian nation in the mid- to late nineteenth century with the Italian (re)-unification, but even to this day, there are vast cultural differences and varying perspectives on ‘national’ identity between regions. For example, a modern day Sicilian, a southern individual from an incredibly diverse culture who speaks a language unlike the italic varieties found in the northern and central regions of the peninsula, is more likely to consider themself as Sicilian than Italian (though they will most likely recognize both ‘nationalities’).

I’m on the move, but a few primary sources that come to mind are several excerpts in the Commedia (i.e., the Divine Comedy), such as Dante’s address to Italy in Purgatorio 6, and Petrarch’s Italia Mia. There’s a lot more information on the topic, but these are some good primary sources for you to get a head start.

The thing is, regional divisions denoting places like Italy already existed in the Roman Empire, and the word "Italia" itself dates to the pre-Roman period. Early in Rome's history, the city-state laying the foundations for its empire would interact with peoples who were called "Italic" by both themselves and the Romans, and who would quickly lend their name to the peninsula. In the late empire the geographic designation would strengthen even further as the official "Province of Italia" would be created under emperor Diocletian in the reforms that would bring an end to the crisis of the third century. While the political history of Italy would see social and political division and turmoil over the life of the two-and-a-half troubled polities which succeeded the Roman Empire, Charlemagne's conquests would lead to the definitive emergence of the term "Regnum Italiae," a construct which would continue to exist through the peninsula's subsequent political divisions.

The development of a unified Italian identity that could couple with the geographic construct, on the other hand, is a more complex topic. During the mid and late Roman Empire, "Roman-ness" had come to mean a legal and social identity as much as it was a notion of geographic identity. While common identities in other European states were, for the most part, built on top of what it had meant to be "Roman" by the outward expansion of a specific region's ruling class who had substituted or fused with that particular region's local patrician class, no such social heartland developed in Italy. Thus in a world where most people's universe is limited to their immediate surroundings (their farm, the local market, the local parish) the absence of a political and social superstratum of people willing and able to develop a unified identity means that this identity probably won't develop, as indeed it didn't in Italy beyond a very small cadre of writers, intellectuals, and aristocrats which emerged sometime in the 13th and 14th centuries, long after more coherent identities had already taken root among the elites of Italy's peers.

I actually wrote this slightly rambling answer to a very similar question on Italian identity awhile back, but if course it is in no way the definitive and final story.

/u/Daeres has previously answered Why do we call Italians Italians? Why the name change, why aren't they called Romans?