I think there are two sides to this question.

The first would be the use of Columbus as a figure of social and cultural shorthand. Indeed, Italian-Americans were not the only nor the first group for whom Columbus represented an important link in the "Foundation Myth," although they definetely enthusiastically embraced and championed Columbus' story and role in the nation-building narrative. Given this function, Columbus' myth didn't even have room to reflect the realities of his actions in the Caribbean, let alone accommodate details of the 15th-century Italian political landscape. You can read more about Columbus' role in the cultural pantheon in this lengthy discussion from about a year ago, with contributions from u/EdHistory101 , u/ /u/Georgy_K_Zhukov , /u/Snapshot52 , and possibly other contributors whom I have missed.

The second point is that even though there was no polity explicitly calling itself "Italy" until 1862, the word was a long-established name for the peninsula. By the time of Columbus' voyage, a shared cultural and social identity had emerged spanning at least as far as the Italian learned class (I wrote about that in this somewhat winding answer from three years ago). It's true that many contemporary sources do refer to him as "Ligurian," although a few also explicitly call him "Genoese" (and at least one specified he was from the ligurian town of Savona, which I mentioned in this answer of mine from a long while back on source material on Columbus) but do keep in mind that most of our contemporary notions of nationhood are actually fairly modern. What this means is that the modern Italian identity emerged in tandem with the 19th century unification movement, and appropriated all of the Italian peninsula's history as its own. I touched upon this idea recently in this thread where I approach the topic of nomenclature and “Italian” identity from the Sicilian and Southern Italian perspective.