Far from it.
The Reign of Italy was born only in 1861.
Before that, for centuries, the Italian peninsula was divided as follows:
A. In the north: a constellation of Duchies (Milan, Mantua, Ferrara, etc.), Reigns (the Savoy), Republics (the Serenissima), Municipalities (hundreds) with completely autonomous jurisdictions, languages, traditions, and oftentimes fighting against each other extremely bloody wars lasting for decades at a time.
B. In the center: the Vatican.
C. In the South: the Kingdom of the two Sicilies.
Once the Savoy unified the Italian peninsula under their helm, one of the Founding Fathers, Massimo D’Azeglio, has been quoted saying: “Now that we made Italy, we need to make the Italians”.
To build a true Italian national identity proved particularly difficult: Venetians had the Serenissima Republic since the Huns invaded the Peninsula one thousand years earlier. In the South there were non-sporadic invasions by the Spanish, Normans and even Arabs, influencing culture, language and also food and it was still vivid even the old Greek culture heritage.
In order to find an example of a truly unified Italian cultural identity, we need in fact to go back to the Roman Empire.
Mussolini swiftly picked up on this idea at the beginning of the XX century with the whole Fascism cultural package and, unfortunately, we all know by now that it was not the most smart or fortunate attempt to re-establish a national identity.
After the disastrous WWII, Italy slowly tried to rebuild its national identity and in my opinion this is still an ongoing exercise which is far from having reached completion.