Empires are often named after the state they originated from. That state may be a nation state or, like Rome, a city state. Like Classical Greece, the Italian peninsula was inhabited by a number of autonomous, self governing city states. The inhabitants of these city states even spoke different languages (eg Etruscan, Latin, Greek).

Rome was one of these city states and they conquered/submitted states around them before moving onto the rest of the world. In the early days of the empire, Roman citizens had a higher level of citizenship than non Roman inhabitants of Italy. So imperial expansion started from Rome- a distinct political entity from the rest of Italy- the capital of the empire and the seat of power.

“Italy” as a Nation is a modern construct. The idea of a single Italian nation was promoted by nationalists to unify the Italian speaking inhabitants of the peninsular in the 1800s into a single Nation. Until then Italy was divided into different states like Venice & Genoa (city states) and the kingdom of Sicily in the south. Hope that helps!