Identity was fragmented in the late medieval period, as it is now. With urban, regional, linguistic, religious, national, and regnal identitarian ties all existing in the same individuals.

National identities which correspond to modern nation-states, such as French or English, existed, and scholars cite to their increasing prominence even in the early 14th century. For example in English royal proclamations which makes appeals and reference to the English nation. So yes, elites had a sense of being ‘Burgundian.’

But on the other hand the rise of mass nationalism is often said to have not matured until the levee en masse of the Napoleonic wars. And you will even find scholarship arguing that France did not truly coalesce into a single national identity until WWI.

The Hundred Years War encompassed a number of regional pushes for greater regional autonomy or independence, which have both important similarities and important difference with modern "independence movements."

Similar, in that they were often delineated along the lines of linguistic or regional identities, like many modern independence movements. France did not speak a unified French but a range of quite different dialects (and cultures), from celtic Brittonic, to Gascon, to Norman. The Gascon-speaking duchy of Gascony fought hard, at times, to remain subjects of the English king. The larger war between the English and French kings encompassed proxy wars between rival claimants in the duchies of Burgundy and Brittany, Brittany in particular was linguistically and culturally distinct from neighboring areas.

But an important difference is that these movements are elite-led to an even greater degree than modern movements; and largely relate to elite dynastic claims and elite desires to lower taxes or decrease feudal obligations, rather than linguistic/cultural/national self-determination for the people.

The life of the average peasant did not change much whether their ultimate sovereign was the French or English king. And laws, taxes, systems of administration were quite localized for the most part. National governance systems, state functions, and centralization were limited. National “independence” would mean little difference except for high ranking nobles (such as the duke of Brittany). This class-based critique is true, to a degree, of modern independence movements. But is far more pronounced in the late medieval period.

The bottom line, is that the idea or paradigm we have of modern nation states, or national identity, existed in a way. But the predominant political identity and idea of how states were organized was a much different feudal one. You are a subject of your sovereign, the sovereign governs by divine right, sovereignty does not flow from the people themselves. This is manifested in how many peasant revolts sought to replace the “king’s wicked advisors” rather than tear down the Monarchical system.

The idea of nation-states, where the state should have a unified language, culture, and governance was largely non-existent. Most political states in Europe were a patchwork of languages, cultures, laws, and feudal relationships. Nobles often spoke different a first language and were culturally distinct from the people they ruled. Though these distinctions were beginning to erode in the late medieval period. Under Edward III parliament was opened in English for the first time and English became the language of courts, but it was not until Henry V that English became the language of the royal Administration.

Identity is of course not monolithic, and there are exceptions and nuance to all of this. But your last paragraph gets the general sense, national identities existed but were much weaker than modern ones, and were not central to contemporary ideas about how states were organized.