I think the best way to say this is sometime between 90 BCE and 48 CE is when Roman and Italian (or rather, multiple Italian) ethnicities became elided.

Dart in his 2014 book on the Social War (and I agree with him) points to the speech the Emperor Claudius gave to the Senate in 48 CE proposing the Gallic elite be admitted to the Senate as a terminus ante quem for this shift. Someone in that speech (different scholars argue over whether it's Claudius or a heckler) draws distinctions between Italian senators (not Roman senators) and Provincial ones, and implies a distinction between the Provinces and... well, someplace that's not made clear as the proper territory to produce a Senate. Here's my translation of those lines:

What? Therefore an Italian senator is not preferable to a provincial one?

it is now the time, Tiberius Caesar Germanicus, that you reveal to the conscript fathers what your speech aims at, for you have come to the far borders of Narbonese Gaul!

The second objection is a bit harder to stretch into an identity claim, but the first one is very clear. Italians should be taken into the Senate, not Gauls.

When this change happened is more difficult to pin down. For a long time scholars have pinned it down somewhere around the time of the Social War(90-88 BCE). The settlement of the Social War included a number of administrative changes that took place over a period of time to adapt the legal structure of Roman government to Italian communities. In 90 and 89 two laws passed that extended the citizenship to all or some Italians (depending on which scholar you read), the Lex Julia and lex Plautia Papiria respectively included the loyal Latins and everyone, or someone, else. There was an attempt at a census in 86 that would have registered these new citizens, but that census was never completed. Sulla is supposed to have endorsed the new arrangement when he returned from the East in 82. In 70 the census finally was completed. During this time Italian communities were reorganized into the little Romes known as municipia, the allied contingents of the army were absorbed into the legions, etc and so forth.

It looks like a neat little package. We have a war, a settlement, a couple of laws, looks like we can wipe the dust off our hands and call the job done, right? Identities elide with the conclusion of the Social War. There are two levels of problems though.

First, there is the question of whether these laws are the result of eliding identities, or the cause. I'm stealing almost all of Mouritsen's analysis of this problem here, and I'll talk more about Mouritsen in a bit. Mommsen, in the 1850s, looked at this bit of Italian history and thought he was seeing a nation trying to emerge in Antiquity. It could never be a modern nation, like his new German nation, because it didn't have representative government (What? Like anyone in the 1850s had a really representative government?), but he saw 1) the Italian demand for citizenship and 2) the attempt to form a state named Italia with a single citizenship as an attempt. Rome dealt with this by becoming that quasi-nation in place of the Italian state and everyone carried on happily as Romans from there (but not in a modern "nation" sense). That position is the one scholars worked with for more than 100 years, with some adjustments. Syme in the 1930s thought mostly along these lines, including the "not-quite-a-modern-nation" bit, but his objection was that Roman citizens existed outside Italy, so Italy couldn't be a nation in the modern sense (as if any nation contains 100% of its citizens and no one else?), and he thought all governments were really oligarchies, so integration only happened when large numbers of Italians entered the Senate under Julius Caesar, but the basic story is the same. Gabba thought, on the other hand, that Italians had already become assimilated before the Social War, and the settlement, the laws and censuses etc., were the recognition of the fact that Romans and Italians had become a single people, instead of those laws and censuses bringing a new people in to the Roman club. Crawford agreed with this position in his chapter in the Cambridge Ancient History on Italians, going as far to say that the army wasn't an engine of assimilation, but rather the fact that Italians could serve in the army with Romans was evidence they were already assimilated. Keaveney in his 1986 book on Italian unification said it was those laws that ended the "Italian Question" Everyone was Roman then, for sure, and that was all there was to think about the matter.

So the idea was that 1) Italians wanted citizenship 2) fought a war to get it and 3) got it, and that was the end of the matter. The questions were when in a narrow period did assimilation happen, and whether the formal changes were the result or the cause of assimilation, but these are really quibbles.

Mouritsen came along in 1998 and said "You guys, you're all being 19th century German nationalists who see nations everywhere, and that's WRONG! If you weren't raised as little nationalists yourselves you'd think entirely differently" His book is really a critique of the scholarship more than an interpretation of Roman history. His main points were that if you're not a nationalist, you'd understand there weren't "Italians" on one side and "Romans" on the other, but instead there were lots of different people on the "Italian" side who had different histories, motivations, and demands. Some of them didn't want the citizenship at all. Some of them thought the citizenship was forced upon them as punishment for fighting Rome, not the goal of the war.

Since then there has been more interest in other ethnicities in 1st century BCE Rome. Bradley argued that "Umbrian" rather than "Italian" popped up as a regional identity to challenge Rome's encroachment on central Italy, Farney's done a study of ethnic difference in domestic Roman politics in this period, Pfeilschifter has an argument about how the army was NEVER a force for integration, but instead reinforced ethnic differences. Dench has done tons of work on Rome as a multi-ethnic society. We're all trying to sort out what the implications of looking at this period without nationalist goggles are and I don't think there's a new consensus except that Mouritsen needs to be listened to, even if you don't agree with him.

For myself, I think for what it's worth, the assimilation of Romans to Italians or the other way, however you want to say that, was driven by the administrative changes that happened in the 1st century BCE into the reign of Augustus, not the other way round. The institutions that modern nation-states have that drive assimilation were really brought together by Augustus, including defining the boundary of Italy, ensuring the franchize for members within Italy, building and maintaining roads (key for modern nation-states), a shared body of literature, etc. and so forth. I think modern nationalism works very well for what is happening in 1st century Italy, which is why I think all those other scholars didn't have problems before Mouritsen called them out (Keaveney even said in the preface to his second edition that even if Mouritsen was right about how nationalist everyone has been, they were still right to be that way). But there are people like Revell, who works on Romanization in the provinces, who says Romanization was really not much to do with people being or acting or thinking like Romans, but instead people telling themselves they had control over their own communities.

So, I guess you can take your pick, depending on what you think political and/or ethnic integration looks like, of several points or arguments or perspectives. I don't think anyone pushes distinct Roman and Italian identities past 48 CE though.