This is a very interesting question!

First, a few notes to serve as a form of disclaimer. The matter of self-identification is inherently tricky. Whether a certain ruler identified himself as a king of the Franks, whether the people of the Frankish kingdoms identified themselves as Franks or thought their neighbors to be Franks, is difficult to dissect, (I will touch a bit more on this at the end). However, what is clear is that the largest division of the Frankish empire occurred with the death of Louis the Pious, who followed the old Frankish traditions of division of the territory amongst his sons. Grandsons of the most famous Frankish ruler, Charlemagne, Lothair I (Middle Francia), Louis the German (East Francia) and Charles the Bald (West Francia) divided the empire up amongst themselves under the Treaty of Verdun (843), bringing about the end of the Carolingian Civil War that had transpired during and following the death of their father.

The Frankish Kingdoms, first the Megovingians and then the Carolingians, had a complicated form of succession and recognition of territory. This often meant that in periods where the ruler had many sons, the kingdom would be divided, and then coalesce together again when one king reigned supreme. The process of inheritance was one that had begun with the earliest Frankish king, Clovis I, who divided his kingdom amongst his sons Theuderic, Chlodomer, Childebert and Clotaire. This partition created the new political units of the Kingdoms of Rheims, Orléans, Paris and Soissons, whose borders and significance would change, although Paris remained a city of great importance. It is apparent that even amongst the earliest Frankish rulers, the ideas of territory, kingship and inheritance were complicated and ever changing.

Despite this pattern of continued fracturing and reformation, the division of Middle, East and Western Francia under the Treaty was final. All three territories developed their own set of rulers and dynastic succession, eventually building in the West to France and in the East to Germany. Middle Fancia, due to its inherently tenuous position between its two larger neighbors, was short lived and divided further at the Treaty of Prüm (after the death of Lothair I). At this point, the common patterns of division and reunification appeared to have a more concrete result. The former territory of Middle Francia continued to be sought after by both its neighbors until the 20th century, where the borders between modern France and modern Germany were decided following World War II.

I will touch briefly upon the matter of self identification, as although I think it is somewhat tangential, I would argue it is an important disclaimer to add to answers to questions such as this. The reality is that it is difficult to ever know who the common citizen would have identified himself with. Would he have recognized himself as a West Francian the moment after the Treaty of Verdun was signed in 843? Likely no. Would he have identified himself as a Frank, and his king as “King of the Franks”? Gregory of Tours, one of our limited sources from this period, seems to have recognized the Franks as a unified group with Clovis and other kings at its helm. But the average citizen may have identified himself as Alemanni, Roman or Goth, or any other myriad of ethnicities that were part of the Merovingian and Carolingian dynasties and territories. Even Gregory, in his heroic retelling of the lives of his predecessors, remarks on the frequent unrest that often came between the Franks.

I am weary of relating the details of the civil wars that mightily plague the nation and kingdom of the Franks; and the worst of it is that we see in them the beginning of that time of woe which the Lord foretold: "Father shall rise against son, son against father, brother against brother, kinsman against kinsman."(Book 5)

So, while the Treaty of Verdun is widely recognized by history as “the moment the Frankish states stopped seeing themselves as part of the same realm”, I would argue that it is difficult to establish when this change was felt by the theoretical common man. Continuous fighting over the territory of Middle Francia may be evidence to the fact that the West and East may have felt as though that territory continued to belong to them, and was wrongfully taken from their grasp. On the other hand, the common man may never have seen himself as a Frank, identifying himself by his family, his neighbors, or the small patch of land he may have called his home.

TL:DR: The Frankish Kingdoms formally "ended" under the Treaty of Verdun, where the Carolingian territory was divided amongst the sons of Louis the Pious. While Frankish tradition held that land was divided among sons, often these territories and kingdoms would reunite and borders would blur. However, the Treaty of Verdun marked the beginning of the legitimate, final separation between East and West. Matters of self-identification are more complicated.

Sources/

Gregory of Tours Translation used: https://sourcebooks.fordham.edu/basis/gregory-hist.asp#book3

Rio, Alice. “High and Low: Ties of Dependence in the Frankish Kingdoms.” Transactions of the Royal Historical Society, vol. 18, 2008, pp. 43–68. JSTOR, www.jstor.org/stable/25593880. Accessed 11 July 2021.

The Frankish Kingdoms under the Carolingians, 751-987 by Rosamond McKitterick

The Penitential State. Authority and Atonement in the Ages of Louis the Pious (814-840) (Cambridge, 2009). Mayke de Jong

(I have plenty more source material related to these kingdoms but not this specific answer if you would like, please let me know!)