Sort of. The city wasn't really part of Latium, it was a predominantly Latin city on the edge of Latium, with a heavy Etruscan and Sabine influence. From an early period the Romans recognized themselves as largely being alike to the Latins, and held hegemony over the Latin League until its dissolution in the Latin War, but well into the historical period they considered themselves to have a sort of unique identity as Romans. Roman colonies were founded in Latium from an early date, and following the end of the Latin War many cities were incorporated directly into the state. The rest were given Latin status, which was a sort of partial citizenship that granted most rights of citizenship except the vote. This was a political status, not a cultural one, and increasingly it became disconnected with Latium as it was granted to other cities throughout Italy and eventually into the provinces--at the same time most cities in Latium were becoming fully-fledged Roman cities. Up until the Social War Latin status was one of many kinds of allied status. Following the Social War most of the socii became citizens, including most of the Latins, by the Lex Julia. After this point Latin status is pretty much purely political, and we see it used in the provinces during the Principate as a sort of stepping-stone for communities before citizenship--already in the late Republic we see this occurring with the Transpadanes, who already were seen as being groomed for citizenship, which Crassus attempted to give them and Caesar finally succeeded in doing.

How much can we say that the Latin identity survived into the late Republic? Hard to say. The Etruscan identity survived into the Principate, even though Etruscan culture had died out centuries before, but the Latins were more connected with the Romans. Conventional logic is that the socii, particularly the Latins who revolted (Venusia), were after Roman citizenship, but there's a lot to challenge that that's been surfacing recently. Most of the Latins didn't revolt, true, but the Latin wolf appears on coinage until quite late, and Latin cults like the cult of Castor and Pollux were always quite popular in Latium. I'm not an expert on this by any means (I do political history), but it's definitely a question under investigation