Only the extreme southern part of Marche speaks a dialect more closely related to the South than to rest of Marche. You can call it "Neapolitan" because Neapolitan is the variety who developed a significant literary standard among those group of vernaculars, but still the spoken language of the southernmost Marche is hardly mutually intelligible with the one spoken in Naples. I know it, since I come from Southern Marche

(not the part speaking Southern dialects though).
Those dialectal areas, however, never played an important role in shaping politics or identity, except maybe in very recent times. No historical Italian polity, with the very partial exceptions of Venice and maybe Genua, ever claimed an area on the basis of the dilectal families. The elites just did not care much of the dialects. When a city was locally prominent for whatever reason, its dialect could have a literary status, so that Italy is plenty of interesting vernacular literatures. Some of them (such as Neapolitan, Romanesco, Milanese and Venetian) have a national prestige, and most Italian dialects, even of restricted ares have established written standards. But dialect FAMILIES, as opposed to local individual dialects, never mattered much to my knoledge.
"Neapolitan" in the sense of the family of Italian southern dialects grouped together is a construction of modern linguistics with little relevance to average people self-perception. I hardly see any political consequence of the dialect spoken in Rome being closer to Naples than to other dialects of Central Italy. As I said, it would affect the literary history and the Italian language to the extent Romanesco idioms leaked into it. The Giocchino Belli analog would probably be less understood throughout Italy and the current standard might be a little more resembling the Southern accents and richer in Southern idioms. This might, in a very minor way, favor the estrangement of some Northeners towards the capital and the South, but I don' t see any major impact.