Hiya!! So not a cultural historian or an early medievalist, but to be honest I would argue the idea of national cultural identity real did not come about till the 1700s or 1800s in either place. Let's consider France (mostly because I am more familiar with it and because it is slightly more organized from a nation state perspective. The modern French that most of us are familiar with was primarily localized around Paris and only became nationalized during the French Revolution. France relied on local legal traditions and a complicated system of precedents and privileges until Napoleon established a uniform legal code. Prior to this period, there was a huge variety of langue d'oil, that were spoken in northern France with Occitan and even Basque variants in the south. This combined with the localization of legal and cultural traditions makes it really hard to say that the overall population living in the kingdom of France had a national cultural identity even if the elite classes were somewhat uniform (they generally weren't in the Middle Ages). Even in terms of religion, French Protestantism was highly concentrated in the southwest of the Kingdom leading up to the French Wars of Religion. The short answer here is that talking about nationalism, national identity, and nation states is incredibly difficult and out of place before the modern period. I guess you could try to argue about the court culture of the elites, but even then I don't think you could apply homogeneity in the Middle Ages and most certainly not a broad level of the underlying populations.

As an additional aside, I may be in the minority, I don't really buy the Treaty of Verdun as a critical point of defining the organization of Europe and its balance of power for the next 1000+ years. I think the historical consensus is definitely taking a more nuanced view of this now, but this still seems to be a fairly common, yet unaddressed trope in popular histories and textbooks. The historiography seems highly presentist, basically projecting French and German borders backwards onto the borders of the treaty and then arguing that most conflicts were inevitably over the geographically doomed territory of Middle Francia. This may be overly simplistic in its description, but I hope you can see how deterministic this idea is. Geoffrey Barraclough was one of the key opponents of this idea and I think there is a lot of room to pushback against the treaty of verdun. Its terms were almost immediately contested by all parties, and despite the idea of rigid structures, various parts of east and west francia had no problem opposing their "king" even after the treaty if it suited them. As a person from an economic and political science background this theory falls further flat still. The Frankish realm would most certainly fall closer to the idea of a conquest state than a nation state, and you can't separate the Treaty of Verdun from the tradition of paritable inheritance (i.e. The kingdom was property of the king to be distributed fairly amongst his sons). Neither region would have extensive state institutions for law, taxation, or military; and at least in the case of France, centralized institutions did not develop in a linear manner, they could be built up under one king to be lost under another. It's also unclear why this would be the perfect "save file" so to speak. There were other diplomatic changes over time, such as Ottakar of Bohemia's ambitions in the east, the Staufens' interest in Sicily, or the treaty of bretigny (which gave the English king the now French province of Aquitaine in full sovereignty). None of these treaties or arrangements stuck, but they were made and contested for long periods of time. I don't want to discount the treaty of verdun completely, but I do want to encourage people to be careful in forging France and Germany in the ninth century. The treat of verdun got some stuff right, but it got a lot wrong, and we should be careful in making it the start point of modern Europe, when the history is so much more complicated than that. Hope this helps and let me know if you have any follow ups!!