There are fundamentally two components to this answer: why medieval Italians called some forms of government "Republic," and why modern readers typically agree with them.
The modern meaning of Republic doesn't really align with the meaning of "Italian City-Republics" (not incidentally the title of a popular reference work on Medieval Italy). Modern "Republican Ideals" are a product of the political philosophy emergent in the second half of the 18th century; these ideas influenced the leaders of the French and American revolutions, who then put into practice government through elected representatives. History has proven this form of political organization infectious, and it has been adopted (with varying forms and with varying success) by most of the world's sovereign states. But these "Republics" as we modern readers understand them would be alien to the politics of Medieval Italy, and even the broadest modern definition of "Republic" would probably disqualify the most inclusive of political organizations in Medieval Italy. For starters, Medieval Italians did not distinguish the body politic and the state itself: the "Comune" (the word Medieval Italians called their "City-Republics") was both the physical and political manifestation of the polity. Thus the state was the people and the people were the state, and in the comune political power was not representative: participation was determined by precedent, privilege, and convenience. In short, far from representative!
Why then, are most people comfortable with the expression "Italian Republics?" This is particularly poignant because the phrase is not without its own risk of confusion: there were some polities in Italy which indeed insisted on the use of the word "Republic" to describe their political system, and while we have also established it is inaccurate to call the comuni Republics as a collective, the practice is nonetheless very common. So when someone is saying "Italian Republics," are they inaccurately referring to all the polities in Medieval Italy, or are they accurately referring to the half dozen polities which, for whatever reason, really liked the idea of the "Res Pubblica" hundreds of years before it would again be unearthed in the "Age of Enlightenment?" We can only look at the context in which the phrase is used in order to draw a conclusion.
The explanation for the continued use of the phrase is actually fairly simple: the phrase "Republics" can be used to easily and unequivocally distinguish political organizations in Italy from the competing form of political organizations in Europe and in Italy.
We can sit for hours and examine the finer points of what or what does not figure in the definition of Feudal Monarchy (and this includes the very value of the phrase "Feudal Monarchy"). But what is clear is that the Italian Republics represented a form of political organization parallel (or even in opposition to) political organization in much of the rest of Western Europe. If, as it has been argued, in most of Europe the post-Roman transformation saw a shift from the "Politics of Empire" to the "Politics of Land," by the late Middle Ages in Italy we have an emergent "Politics of People." Sure, semi-autonomous urban communities existed all over the Mediterranean Sea and the rest of Europe, but only in Italy were they the most common form of political organization.
If in the rest of Europe the state was to be personified in a monarch, which however limited by powerful aristocrats or constrained by legal precedents, nonetheless governed by divine right (although these precise notions would not be laid out until some time after the medieval era) in Italy we have on the one hand the Germanic Kaiser who through a convoluted accident of history appropriates the language and constructs of government as "Empire," and on the other a diametrically opposed and entirely organic development whereby urban communities derive their autonomy by the mere fact that they exist.
Thus the Italian comuni could not be monarchies because they did not derive their authority from a social contract between monarch and populace (and I am aware the notion of social contract would be defined long after the middle ages; I'm merely co-opting the concept for clarity). The governed and the government, while not one and the same, were not clearly defined in the Italian comuni. Political representatives, and political appointees, swirled in a cauldron of overlapping and continuously redefined spheres of influence. This was not a game for the faint of heart, with consequences of political defeat ranging from loss of property to loss of life, in part because of political responsibilities and political factions are unequivocally linked to deeply personal needs and responsibilities.
Starting in the early thirteenth century, some comuni saw charismatic politicians climb to positions of authority and dismantle collegiate institutions, expelling their rivals and governing unopposed. But here too no notion of divine right emerged: succession could zig-zag across members of a political dynasty, with authority jumping between brothers, cousins, and even extended family or friends as individual members of a dynasty on the one hand jockeyed for political appointments and influence, and on the other tried to guarantee successful government as well. A sequence of bad decisions could spell the end of popular support for a previously unopposed dynasty, however on the other extreme a particularly successful or popular figure could outright govern without a formal appointment within one of the comune's institutions. The people, not the monarch, ultimately defined government the Italian City Republics.