So, Northern Italy and Southern Italy history diverged a lot since the destruction of the Lombard Kingdom. They were politically split since IX century to XIX century, more than a thousand years. Still, culturally, they didn't diverged a lot. While they have some minor differences, it's not the ones you expect in a place with that cultural influences. In the north, they were conquered by french, german and spanish mainly, in the south you have arab, greek, norman, spanish (both aragonese and castilian), german and french rules. Also, for long periods, they remained as free minor states. And, after all of that, they were all considered italians. You have examples of countries that started with some culture but split, like the slavs on the former Yugoslavia who at the VIII century were all just slavs. But that didn't happened in Italy.
So, how did the Italian culture not split between south and north, despite the culturally diverse conquerors and being split politically for more than 1000 years?
How would you define "split up?"
Very generally, modern readers can sometimes underestimate the impact of relatively recent institutions in modern unitary states and how they shape national cultures. These institutions creating a common identity can both be public, like national social services or national school curricula, as well as private, which includes things like nationwide businesses, national newspapers, and especially radio and television. And many of these features that modern readers would list as features of national identity and culture were in fact introduced as late as the 19th century, which was a period when European social and political leaders paid particular attention the narrative of national identity and national culture. Prior, notions of identity were still mobile and could be influenced by things as disparate as populist moods and sentiments, all the way to the imposition of what a small cadre of politically active intellectuals believed defined a shared identity.
In some places, like France, efforts to impose a unified national identity to the detriment of local identities was a hallmark of 19th century policy-making. In other countries, like Spain, efforts were less deliberate until the following century and the consequences of these cultural policies is still now ambiguous. But what is consistent regarding identity in Europe is that it difficult to extrapolate modern notions of identity prior to industrialization, and it is equally difficult to compare notions of identity across modern borders. In the 17th century, would the differences in habits and language separating a Venetian with a Neapolitan be more marked than the differences between a Parisian and a Marseillaise? What about the differences between the average resident of Madrid and Barcelona? Even among the upper classes, there is no guarantee that they would have been tutored using the same texts, or consider the same works cultural "classics." In an age before refrigeration, would anyone have been able to eat the same foods? In an era before mass media, would people have been interested in the same news? Just as an example, the city of Nice was majority Italian-speaking and had ecven spawned some of the key activists in the unification process, until the Kingdom of Piedmont-Sardinia traded it to Napoleon III's France in exchange for military aid against the Austrian Empire. Decades of policies targeting cultural assimilation would eventually turn Nice into a French-speaking city. However, the Alpine region of Savoy, also traded to France by the Piedmontese, was much more easily integrated into the French cultural fabric (whatever that might mean). Sure, Savoy was much more rural, less densely populated, and tucked away in the Alps, thus much different from Nice, a bustling maritime city. But the point is that cultural homogenization was a complex process, and that process took different shapes in different places.
Further, prior to the emergence of modern notions of nationalism identity itself could exist on a spectrum. For centuries, communities of Italian speakers existed on the Dalmatian coast and on Aegean Islands. As an example, Ugo Foscolo, a poet and intellectual active just prior to the unification process, was born on the Agean island of Zakinthos which Italian speakers like himself called Zacinto. Foscolo could read and write in Greek, but strongly identified solely as Italian. Others in his same community on Zakinthos might be more comfortable speaking in Greek and identify as such, while still others might consider themselves somewhere in between, or not give the topic much thought at all. While I have cited a single example, on mainland Italy and even the rest of Europe too, people could have different identities that they emphasized or de-emphasized in different situations, or depending on their social upbringing, and even personal preferences. All this colludes to make pre-industrial notions of cultural and social identity very difficult to assess.
In Italy, efforts to create a unified national identity were a serious preoccupation of the political and intellectual class in the second half of the 19th century. The young country's debate on a standard language and school curriculum is particularly telling, as discussion on language standardization had existed in the intellectual space for nearly four centuries prior to unity. I wrote an answer the standardization of Italian here a couple of years ago which might help you get an idea of the kind of hurdles that the nascent Italy faced in constructing a common identity. Matters were further complicated by the ambitions of the northwestern political-intellectual class driving the unification process in that they had initially not even included the southern part of the peninsula in their plans. Indeed, the conquest of the south was largely catalyzed by semi-autonomous activism by the immensely popular Giuseppe Garibaldi, a fairly radical actor in the landscape of the Italian Unification process. After Garibaldi's conquest, the cultural and social differences between north and south would hamper effective policy-making well into the 20th century, and lead to the emergence of a wide divide in terms of social and economic development separating the two halves of the country (a wide divide which not only persists to this day, but has been widening).
I had examined the preconditions for Northwestern Italy's role as catalyst for the unification process and the correlated lopsided industrial development of the peninsula in this admittedly very long answer, also from a couple of years ago, which examines the drivers of a phenomenon known as "Italian Dualism." The contemporary manifestation sees Italy's northern regions align themselves in terms of economy and society with European averages (with some regions, like the provinces of Milan and Bolzano, widely surpassing EU averages) while Italy's southern regions exhibit developmental characteristics well below the European average (some regions, like the isolated southern region of Calabria, are almost half as developed as the EU average).
While you might not have the patience to dig through my five-part answer (and I wouldn't blame you, in retrospect I definitely should have trimmed it) the main thesis is that while pre-unitary economic development was only slightly better off in the north at the time of unification, the social and political conditions in the south meant that the North would capture the vast majority of Italy's industrial and post-industrial growth. Thus we can say that there were indeed deep differences between the two halves of the country.
While my linked posts focus on economics (I am, by training, an economic historian) there is also something to be said of Italy's north-south dualism in terms of social and political expressions as well. In the Italian election of 1992, the influential "Northern League" party emerged pandering to Northern Italian's grievances against the South and remains a very influential political block to this day. While in recent years efforts have been made to give the Northern League (re-branded as "the League") a national appeal, these events break our twenty-year rule. I would point you instead to this Tuesday Trivia post from about a year ago that I shared explaining some aspects of the Northern League's emergence in the early 90s.
Lastly, I would conclude by adding that Italian media has also exhibited regional quirks, revealing that the country is much less homogeneous than might be implied in the question asked. Historically, Italian media was dominated by the state broadcaster RAI. Italian governments had a habit of using state-owned corporations to stimulate the disadvantaged south, and RAI was no different. RAI typically hired southern talent and thus appealed to Southern viewers, building a narrative that saw southern culture and mannerisms become the representative standard in Italian media (this also greatly seeped into media that Italy exported, much more likely to focus on Rome or Southern cities than anywhere in the North). Indeed, part of the runaway success of the Mediaset media conglomerate in the 1980s was its ability to pander to northern viewers within a context that saw Milan become the cultural reference point in Italian media instead of Rome. I wrote about this phenomenon in this answer here.