I think people are overstating the Mediterranean as a border and making things difficult. Its not. If it wasn't for the islamic conquests dividing the Mediterranean in half, then we likely would associate the whole sea with one region and northern europe with another. More over, trade and connections tied these people together long before Rome came about and long after Rome fell.
The big thing is legitimacy, successive empires or republics or what have you, uniting the region and being to claim that they are rome and for others to respect that claim. That is really what matters. In regards to Identity I can easily see where over time the Mediterranean basin considers themselves roman but with many other ethnic groups scattered about. Just like how in China today you have the Miao, Yi, Yue, Zhuang, Hui, Tujia, Kam, Bouyei, Yao, and Bai, just to name some of the many ethnic groups in China.
In regards to language, the main difference between China and Rome isn't the spoken language but the written language. There are man Sinetic languages in China, Manderian, Cantonese, Wu, Min, Jin, Xiang, Dungan just to name several of the languages. However till the modern era they generally used the same writing system. the easiest way I can see to have something like that is to have the romans adopt the greek script. This also would help bind the empire together more, thus helping overall.
When it comes to clothing, that also evolved. We today might call it all hanfu, but to say that there weren't differences over time is silly when the records show otherwise. fashion historians can identify the dynasty just from the dress. Likewise clothing of the various eras of Rome changed, the various eras of the Byzantine period share a commonality with classical garb. So I can see an evolution continuing for the rest of the empire if it managed to rebound and survive. How it would evolve I am not sure, but we might see more similarities, perhaps trousers never really become commonplace for the roman citizens? Hard to say really.