I'm afraid /u/erissay makes a number of mistaken, if not outright wrong, statements.
Almost every scholar from Mommsen to now, when writing about Italy of the 1st century BCE, described it as a nation-state (or synonymous terms) except in one way. Mommsen said Italy approached a modern nation-state (I'm not sure what the German was - the translation I have gave the synonymous term "nation") except that it lacked a truly representative legislature. Syme said Italy might as well be called a nation except the citizenship extended beyond the borders of Italy. Keaveney basically thought Italy in the 1st century was analogous to revolutionary France. Indeed, Mouritsen's critique of the entire scholarly tradition about the Social War was primarily that everyone conceived, consciously or unconsciously of Italy as a nation-state.
There are a couple more-or-less contemporary theorists of nation-states that also argue for ancient nation-states in a not primordial way ("primordial" being the idea of nation that claims an essential, national quality that unites all members of the nation across all time - so that the ancient Germanic tribes that fought Rome were the same as the Germans of the 3rd Reich). Smith argues that ethnic groups ("ethnie" - I'm not sure why he uses the French word) can indeed have long historical spans, with strong linguistic bonds, a continuous history, a shared memory of a homeland, and continuous traditions (such as liturgical languages and practices), and these ethnie can from time to time become politically galvanized, so that a national movement arises, and when these movements are successful they are nation-states. That's not to say it's the same nation-state reemerging over and over - he's not a primordialist. He does however see, say, ancient Judea or ancient Armenia as true nation-states. He argues about all those "except this one thing" in the scholars I already mentioned: if your measure of a nation-state is a truly representative legislature, it seems ridiculous to argue that any such thing existed before Women's Suffrage. How could a legislature that represents only half the population be truly representative? If your measure is the nation-state contains all its citizens, what state today does?
Italy was absolutely unified as a political entity that was not a province by the reign of Augustus, probably in 42 BCE, where Italy is defined as the peninsula between the Gulf of Tarentum and the Alps. All communities were joined in a single citizenship with shared rights and responsibility to the state, and although that citizenship was called "Roman" we see e.g. the emperor Claudius in 48 CE addressing the Senate as the Senate of Italy, not the Senate of Rome. Administration of the state was conducted at Rome, but included Italians at all levels. Some scholars argue that Italy was conceptually unified as early as the Gallic invasion of 222. There is a nice discussion in Williams, Beyond the Rubion, on the development of a unified Italy and different views on it.
Italy was never called "the Roman kingdom," unless that's not what they mean by
Pre-Empire, the area now known as Italy was slowly conquered by the Roman Kingdom and was thus called as such.
I've already discussed how Italy did exist as a distinct geographical and administrative entity, and a conceptual entity at least before that. By the 1st century BCE AT LATEST Italy and Rome had been joined in a single entity managing the empire.
As to why it's referred to as the Roman empire, we might be informed by the names Romans (including Italian Romans) called it:
Imperii nostri - our empire. Imperium Romanorum - the empire of the Romans. Romani imperii - The Roman empire. They are not talking about "the empire of the people who live in Rome;" they're talking about "the empire of the Romans," which by the 1st century BCE included the Italians.