Well, there are two complementary ways of addressing the question.

On the one hand, simply because it's a name and that's just how names for things work. We don't call it Modern Latin 'because' we call it Italian. Similarly, we distinguish between dialects and languages with some variation, but it's a judgment-call.

The second reason, the other hand, has to do with the lines of descent for those languages.

In the case of English, going backwards, we use the terms: Modern English > Early Modern English > Middle English > Old English (I'll stop there for now)

The only other offshoot from Old English that gets much discussed is Scots.

Latin, on the other hand, has two issues that play against Italian being called 'Modern Italian'. Firstly, let's work back from Italian.

Italian > Tuscan > Italo-Romance > Vulgar Latin // Latin.

The issue is when you go the other way, the number of Romance languages that spring from Vulgar Latin is numerous and broad. Why would Modern Italian, the derivation of a singular subset of that heritage, have more of a claim to being called 'Modern Latin' than Spanish, Portuguese, Romanian, French, or dozens of other, smaller language groups?

Secondly, the history of Latin continued parallel to the evolution of Vulgar Latin into the Romance Languages, and Latin exists today in usage (admittedly relatively restricted usage and it is hard to make an argument that it's 'alive', but it's not 'dead' either. So there is already a language, neo-Latin, that occupies the conceptual slot of 'Modern Latin' into which you think Italian should go.

Those two factors, the non-unique claim of Italian as a successor language, and the ongoing usage of Latin, compared with English, which didn't have that many offshoots, and doesn't have an ongoing 'competitor' language for the same terminology, seem good reasons for not classifying Italian as 'Modern Latin.'