Given the lack of answer this question has, I'll try to answer it as best as I can, considering that I am only a student, but that I did actually study extensively the Hundred Years War in France and hope I'll be somewhat up to standards. My main source will be Boris Bove "Le Temps de la Guerre de Cent Ans"

You question can be divided into two, because while there was indeed a sense of French nationalism, albeit a small one, it was not the main reason that made people accuse Burdundians to be false French.

First of, French Nationalism. Obviously, nationalism back then was nowhere near as important than what we have today, but nationalism was present back then. More especially, it's pretty much the time where nationalism asserted itself and became an important component of French Political life : if anything the history books tells us is that war, especially on such a long period, is a great stimulant for nationalism. It's even more the case for the Hundred Years War where the opponent was not exactly culturaly distinct and actively claiming both thrones : that embryo of nationalism was a way to prevent that, to distinct the two entities, to culturally drift from the common identity that could be the basis of unification of France and England under the English King.

But we can go back sooner in time to see evidence of French nationalism, with the selection of Philippe VI of Valois rather than Edward III, who would have a claim through his mother. The only real reason that Philippe was choosen, aside from his already rather powerful position in french politics, is the fact that he was not English.

So yes, there was already nationalism, obviously not as important as today, but still important enough and nationalism would only grow throughout the century as a natural consequence of years of warfare that directly impacted the populace (the english relied mainly on raids and pillage throughout the countryside in the first phases of the War).

The hate against Burgundians is different however, because it's more of a political than a cultural thing.

After a few wars that one went one way then the other, Charles VI ascended to the throne and while his first years looked promising, he eventually turned mad. While everyone thought that he would eventually die quickly, he actualy did not, he reigned a whole 42 years, 1 month and 5 days, while alterning between dementia and lucidity. Such a situation will eventually require a regency with at its head his brother, Louis of Orléans, a very powerful man at the time who kept with the centralising and administrative expansion policies of the previous kings. Such power will eventually raise jalousy, espencially from Jean the Fearless, Duke of Burgundy, at the head of an impressive amount of land and whose political ideas is much more about decentralisation and a return to the "good old times" so to speak, to return as things were under Saint Louis, basically the opposite of Louis's ideas.

The political and personal spat between the two quickly turned central in french politics, view this almost like modern-day political parties, both sides the Armagnacs (Louis) and Bourguignons (Jean) had their own emblem that some people would publically wear etc.

Eventualy, all this would lead to the murder of Louis d'Orléans, murder for which Jean did not even apologise, but actualy claimed the murder and justified it as the removal of a tirant. From then on, civil war was pretty much inevitable (which explains the British victory at that time), with the Bourguignons supporting the British while the future king of France, fled south to Bourges with the rest of the Armagnacs.

All this, political differences, personnal rivalries and later on treason toward the rightful heir is much more the reason for the mention of them as "False French" than anything. It is true that the Duke of Burgundy had several lands outside the Kingdom of France. It is also true that the Duke of Burgundy, as a proponent of decentralisation and not personnaly concerned to the fate of France, probably considered themselves first as Bouguignons than as French, but it's much more treason and overall animosity with the Armagnacs and supporters of the rightful heir that gave them the name rather than nationalism.

Finaly, I would underline the fact that Bouguignon is much more a term used as a political party rather than the cultural identity here, the insult is aimed at the followers of the traitorous Duke of Burgundy, not to the random peaseant living 20 miles outside Dijon. The local population was culturally not that different than the population that would be found near Paris, and would have been much more akin to it than the populations of southern France who still had their own language, customs etc.

Hope that covers your question and retains the academic levels required, I'll be happy to provide answer to any further question or make any of my point clearer!