I can't speak to the circumstances in the movie, because I haven't seen it. Pompeii had fought in the Social War against Rome in the early 1st century BC, and subsequently been made a Roman colony (which would essentially mean that the free inhabitants were legally Roman citizens). This means that by the time of the eruption (ca. 150 years later), the locals had been "Roman" throughout living history, but dig back a bit in the history and they had fought against Rome. Interestingly, one of the things the Italian allies fought Rome for in the social war was the right to become Roman citizens, which makes this all a bit difficult, I think, to comprehend.

At roughly this timeframe, we certainly see a sense in which Roman citizens from outside the city identified with their place of origin in particular; the hometowns of poets laid a special claim to that status and identified the author as their own (Martial: Tantum magna suo debet Verona Catullo / quantum parva suo Mantua Vergilio -- great Verona owes as much to its Catullus as little Mantua to its Vergil; this makes my point incidentally rather than explicitly, as really it is a compliment to Catullus more than anything), but that need not mean that Catullus felt himself less of a Roman in his identifying with his Northern Italian roots. Ovid was a native of Sulmo, but the pain of exile evoked by the Tristia is his detachment from Rome. The complication may lie in the double identity of Rome as both a city and a proto-nation.

There may be something of a parallel to citizens/subjects of the British Empire abroad, who could perhaps identify both with their regional identity (and they may well have been born and raised there and never seen Britain itself) and with their overarching British heritage, but I venture well beyond my area of expertise here. Certainly Pompeians might identify in opposition to the residents of the city of Rome, whom they would have seen plenty of as they came down to the Bay of Naples, much like New Englanders will complain about the city people who come up to vacation in the autumn, but if the sense within the movie is that the Pompeians are identifying against the broader sense of Roman citizenship, that would certainly be pushing things; in that sense, Roman-ness had overwhelmingly dominated the culture, architecture, language, and politics of their own city for as long and longer than any of the residents had been alive.

I have not seen the movie so I can't comment on how it is wrong (no doubt it is wrong, Hollywood movies always are). I can give you some background information so you can make your own decision if what you've seen fits what is known.

Pompeii came under Rome's influence during the Samnite Wars, around 300 B.C. For the next two centuries Pompeii, like many other Italian communities, was Rome's ally, providing auxilliary forces for the Roman legions. During this time Pompeii was largely independent in how it governed itself, as long as it honoured the treaty obligations towards Rome. The Italian allies however became disgruntled about the heavy demands put by Rome on their manpower, and the relatively little benefits they saw in return. In 90 B.C. many allies, including Pompeii, turned against Rome. This was a messy war and a generally confusing period, with the Social War (that's the one with the allies) followed by a series of civil wars between Romans.

What matters for this topic though is the outcome:

• After the Social War, all Italian communities received Roman citizenship. From this moment on, all Pompeiians were Romans and the Pompeii community adopted Roman style public offices.
• In the civil war following the Social War, Pompeii supported the losing side. As punishment, Sulla (the winner) gave 2000 to 3000 of his discharged soldiers property and land in Pompeii.

The settlement of a few thousand Roman colonists in a population of about ten thousand led to civic strife and displacement of some of the original population. Within a generation, the pre-Roman elite was replaced by colonists. (Lomas, p. 212)

So more than a full century before the disaster at Pompeii, the city was fully Romanized. The entire population had received Roman citizenship and the city's original elite was replaced by Romans.

Source:

• Kathryn Lomas, Italy during the Roman Republic, 338-31 B.C. (in: The Cambridge Companion to the Roman Republic, edited by Harriet I. Flower (New York, 2004), p. 199-224)

The presentation in the movie is wrong...clearly, the relationship between Pompeii and Rome was complicated in the Social War (as the other comments have shown very well), but the idea that they were actively fighting to be considered Pompeiians instead of Romans, especially as late as 79 CE, is absurd. There is nothing about this in the extant historians, and I have not yet found any similar sentiment in the graffiti I've encountered.

It's true that the Pompeiians had their own identity and connection to their region...you can see this in the fact that there is graffiti in a variety of Italic languages, such as Oscan (a language besides Latin spoken in Italy) in prominent places throughout the city. They expressed their own distinctive culture, and were proud of their heritage. And this is true throughout the Roman empire...you find a blend of cultures everywhere, since Rome allowed people to keep most of their customs, and even adopted some of them (i.e. the Cult of Isis, the use of eastern spices, Greek influences in art). However, in Pompeii in 79 CE, there was not a movement to effectively secede from the Roman Empire, as the movie implies.

EDIT: And can I just gripe that their presentation of the eruption of Vesuvius was totally wrong from a volcano science standpoint? Usually I don't care about such things, but I have spent so much time trying to understand how Pompeii actually went down that I kinda want to show off.