The cheeky answer is that the rest of Europe didn't have a "Feudal System" either (other equally cheeky answers that are also correct include: (1) feudalism wasn't a system, (2) the system wasn't feudalism, (3) the very notion of feudalism is a product of our imagination, and lastly, (4) why do we need to call anything feudalism in the first place?).

But without being cheeky, I'm pretty sure I understand what you're asking: why didn't the Italian peninsula develop a monarchy exhibiting some or most of the characteristics we ascribe to "Feudal" Europe (or why didn't Italy develop a unified monarchy at all, for that matter).

The answer, on the one hand, has to do with the absence of mechanisms which led to these institutions to emerge in the rest of Europe: as politics in the Late Empire were increasingly a power struggle between generals stationed on the frontier, as the rubber band holding the Frontier and Rome together stretched to the point of snapping individual warlords were able to appropriate the local institutional framework and consolidate monarchical rule in those frontier territories. And while grants of land ("Feudalism!") are an easy substitute for salary payment to loyal soldiers once imperial tax revenues disappear, there is a lot of the old roman system which still manages to get transmuted through the centuries (for centuries the Monarchs of Europe still struggled to reduce the influence of the Roman Church, with the German "Emperor" literally traveling to Rome to be crowned for nearly a thousand years).

On the other hand, the Italians would insist on the pageantry and framework of the Empire long after it was sensible to do so. Part of this was inertia: Italy never suffered a total shift in power or governmental structures as the frontier did. Part of this might also be cynicism, as the Italian governing class (especially ecclesiastics) clung to the skeleton of whatever pan-European power framework they could keep propped up. I wrote this answer to a similar question asking why the city, and not the monarchy ("feudal" or not) became the building block of Medieval Italian political order.

One thing that I often point out is that the peninsula was actually a unitary polity for about a century after the traditional date assigned to the "Fall" of the Roman Empire, until it suffered a destructive and destabilizing conflict through which the Eastern Empire (but was it really just the "Roman Empire"?) attempted to reassert authority over the peninsula, strongly reducing the chance of any sort of stable unitary political union. Three subsequent political orders (Lombard, Carolingian, and the earliest Holy Roman Empire) also proved unable to create some semblance of a lasting unitary political system.

But looked at from another lens, if the post-roman centuries could generate three political orders calling themselves "Kingdom of Italy" with very few of the classic signs of "feudalism," might we have to redefine what "feudalism" means in the first place? There's a section of the FAQs on this which might be worth looking at.

One of the things that medievalists will commonly point out is that the sort of Vassal-Lord relationship we have come to expect in the "Feudal System" was not as well-defined as might be expected, nor was it the only sort of relationship governing the medieval social order. But this doesn't mean it was absent entirely, even where it was particularly weak: grands of land linked to political privileges and responsibilities very much existed in Medieval Italy. The defining difference would be that in Italy, the top-down anointment of these privileges didn't really exist (landholders had to justify their privileges to each other, rather than having them handed down by some higher authority - except in those instances where a regional authority figure did emerge, complicating things further) and more critically, there existed parallel, and competing, social privileges and relationships which could be just as strong (if not stronger).

So, I just wanted to add a bit of nuance to u/alvisefalier's excellent response to this question. I would like to thank them first for setting laying out the whole issue with the term feudalism, etc.

I think it is worthwhile to mention, however, to mention a couple of clarifying points regarding this. First of all, you do see "feudal-style"--bearing in mind all requisite caveats when using this term--landed nobles at work in Italy, particularly in the far north. For example, the marquises of Finarius were frequently obstacles to the machinations of Genoese hegemony over the western Riviera.

Furthermore, staying in Genoa--i did my dissertation on this city, but this sort of thing happened elsewhere--politicians in the city states often acted as feudal lords outside of the city walls while maintaining a complicated set of citizen rights within the walls of the city. For centuries, the Fieschi were the most powerful familial faction within Genoa, and they drew their wealth and authority from their estates around Lavagna. Indeed, within that city, you have a two-tiered political class of nobiles--landed nobles with estates outside of the city who also invested in trade--and mercantes--individuals who drew their wealth and power exclusively from trade. This second group was not able to overcome the former until the insanity that is 14th century Genoa when they made a more assertive compact with the lower orders of the city.

The presence of a landholding class was not limited to Liguria in Italy. As Philip Jones describes in From Commune to Signoria--still the best synthesis of political, social, and economic histories of medieval Italy--during the tenth and eleventh centuries, we see a process of "feudalization" (his words, though other Italian historians have used them) wherein larger landholders begin to gobble up smaller estates and establish fairly harsh legal conditions for the cultivators living there. I think this is a point that needs stating because medieval Italy is seen as a land of city states, and in many ways it was, the populations of these cities would not have been more than 30 percent of the overall population--40 percent if we're being generous.

Now, what is my overall point? Basically, I want to emphasize that while there is no unitary system that governs western Europe during the Middle Ages, there were shared social, legal, and political traditions, and if you look closely you see that Italy is not as different as you might first assume.

The high medieval socio-political order has clear roots to the immediate post-Roman period, but we can see some of the clearest trends emerge in the post-Carolingian era, when the various successor granted away lands and titles and the old kingdoms disintegrated. Italy is no different in this respect. By the early/mid-ninth century, the Kingdom of Italy was a hot mess. The father-son royal duo Berengar II and Wido were forced to grant charters to the large cities of northern Italy, granting their independence. That said, these charters contained the much of the same boilerplate legalese that you find in personal charters to some sort of landed aristocrats, granting the same sorts of rights that you would see elsewhere in Europe.

This is all a roundabout way of saying that yes, there are different things going on in ITaly, much of this has to do with the persistence of Roman (and pre-Roman) urban settlement patters, but despite the immense variability of medieval western Europe, patterns emerge that connect Italy with the rest of Europe.