As for who the Bastard of Polignac was? After searching absolutely everywhere for his name (not just Google, but also Google Scholar, JSTOR, my academic books, everywhere) his name only shows up when that specific event is talked about. Lindsay C. Hurst's "Porphyria revisited" (link below) mentions that he was a chevalier, and a famous one at that - but doesn't go into why he was famous nor why he was a bastard in the first place. Only that Charles VI did kill him in that bout of insanity.
It should be noted that there are two places called Polignac in France. One in Haute-Loire (central-southern part, Auvergne-Rhône-Alpes) and one in the Charente-Maritime (western coastline, Nouvelle-Aquitaine). Besides, the only records I could find of a Duke of Polignac is from the 1780s onwards, when that title was created.
As for why his name survived - we need to look at the bigger picture:
Polignac was a chevalier during a time where the knights and nobility still had significant sway in the French political system; even as they had suffered significant setbacks earlier in the 14th century - not just events like the Great Famine and the Black Death, but also the then ongoing Hundred Years War demanded a lot of attention as well. The most powerful of nobles, like the Dukes of Burgundy and Armagnac, controlled sizable territories on their own accord and threw their weight in the war as well on either side - making their support very important for both the English and the French kings.
In other words, France during the Hundred Years' War was a very tense place, even during (relative) peacetime. While 1392 saw no action between the English and the French, Charles himself had already had several events go against him.
First, his Constable of France - one of his closest allies - was almost assassinated in June 1392 by Pierre de Craon, a former chamberlain. The would-be assassin sought refuge with the Duke of Brittany, and Charles - considering this event an act of treason - raised an army to invade.
Oh, and the campaign is where he first went mad; in front of the entire army, forcing them to abort the campaign completely. The primary sources differ in the details, but they're unanimous on the event itself taking place generally.
After he returned to Le Mans, the primary sources all mention one event. It is a retrospective attempt to find unheeded warnings, Pfau argues in her doctoral dissertation, but it's still worth mentioning: As he rode past a leper hospital, Charles heard prophesies coming from a leper that the King should beware of treason within his ranks.
And here we go, the event you're asking about. The King, now looking for treason anywhere he can, proceeds to take the first opportunity given to him - one of his men-at-arms accidentally dropped his sword. With that opportunity, the King picks it up and kills four of them before the sword breaks. One of them, although not mentioned by some primary sources, is the Bastard of Polignac.
Again, the primary sources differ in the details, but the result is the same: A king killing a subservient soldiers because he went mad meant that he could not be counted on leading his kingdom. The primary sources describe the illness - and the attack - as a very harmful event for the entire nation. It wouldn't be the last time the King went ragingly mad, but in the context of nation building, the event required the entire Kingdom of France to step forward and support him so he could recover from insanity.
(EDIT 2) I should add that one of Charles' uncles, Duke Philip of Burgundy was present at the Le Mans campaign. And when Charles went insane, Philip took the chance and assumed command - appointing himself Regent of France. Charles' brother, Louis of Orleans, was not exactly pleased with that development, so a 12 year power struggle over the true control over the kingdom began while Charles spent most of that time, well, mad and practically incapacitated.
And that didn't stop with Philip's death in 1404 - Philip's son, John of Burgundy, resumed the fight with Louis; this time over the royal children. And with Louis being assassinated in 1407, things spiraled further and further out of control with an insane king and renewed English aggression (with the Burgundians on the English side). Something only reversed when Joan of Arc got herself into the fray.
To return to the question: it's... really hard to tell. The Bastard's mentioned, but the knight's, again, only mentioned in connection with the King - and thus, only as one of the major catalysts of a major crisis. Again, there's no mention to if the Bastard has any connection to the 1780s and later dukes of Polignac in any manner, so we don't have any family history to go on.
Thus, I'm afraid a lot of the answer has been lost to the sands of time. But the parts of the answer that remain still are important, since the events during Charles VI's reign led to great instability - eventually, Joan of Arc.
Sources: Lindsey C. Hurst, Porphyria revisited, Medical History, 1982 University of Cambridge, 26: 179-182
Aleksandra N. Pfau: Madness in the Realm: Narratives of Mental Illness in Late Medieval France, a dissertation submitted in partial fulfillment of the requirement for the degree of Doctor of Philosophy (History) 2008 University of Michigan (Discusses the primary sources extensively)