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The wiki entry is misleading. Ruth Putnam gives what seems to be the actual explanation in a well sourced ebook, "Charles the Bold, Last Duke of Burgundy":
Separated from this group by a little space at the very edge of the pool, was another naked body in still more doleful plight. The face was disfigured beyond all semblance of what it might have been in life. One cheek was bitten by wolves, one was imbedded in the frozen slime. Yet there was evidence on the poor forsaken remains that convinced the searchers that this was indeed the mortal part of the great duke. Two wounds from a pick and a blow above the ear—inflicted by "one named Humbert"—showed how death had been caused. The missing teeth corresponded to those lost by Charles, there was a scar just where he had received his wound at Montl'héry, the finger nails were long like his, a wound on the shoulder, a fistula on the groin, and an ingrowing nail were additional marks of identification, — six definite proofs in all. Among those who gazed at this wretched sight, on that January morning, were men intimately acquainted with the duke's person.
"There were his physician, a Portuguese named Mathieu, and his valets, besides Olivier de la Marche [18] and Denys his chaplain who were taken thither and there was no doubt that he was dead. It [page 450] has not yet been decided where he will be buried, and to know it better it [the body] has been bathed in warm water and good wine and cleansed. In that state it was recognisable by all who had previously seen and known him. The page who had given the information was taken to the king. Had it not been for him it would never have been known what had become of him considering the state and the place where he was found." [19]
The two footnotes read:
[Footnote 18: It is strange that La Marche does not make more of this scene if he were really there. His sole statement is: "The duke remained dead on the field of battle, stretched out like the poorest man in the world and I was taken and others." iii., 240.]
[Footnote 19: La déconfiture de Monseigneur de Bourgogne faite par Monseigneur de Lorraine. Comines-Lenglet, iii., 493.
This brief account was drawn up evidently before the duke's burial was known by the writer. It may have been written solely to please Louis XI. Still there is a simplicity about it that holds the attention, in spite of the fact that the story is not accepted by critical historians.]
In other words, the giveaway wasn't so much the fact the nails were unusually long, as it was the fact that there was an ingrowing nail. Perhaps more importantly though is what footnote 19 raises: the contemporary description of the body being found and what happened next may have been written to please Charles' rival, Louis XI of France -- and the story's veracity is not taken for granted by historians.
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Dont have an answer, only an opinion and a small piece of evidence, but it is too long for a comment.
Opinion: Maybe they did not stand out so much. The text may mean that the nails were one piece of evidence, not a decisive one. Should we suppose that his hair color were so unique, too? The last one, "old battle scars", appears to be the most distinctive.
Weak evidence: in his previous defeats at Morat and Grandson, the swiss had looted his personal belongings, even his personal war tent with clothes and his personal seal, which is in Zurich's landesmuseum. In the museum it is even stated that he had difficulty to issue orders as all orders were supposed to be authenticated by the seal.
It is thus reasonable to expect him to be wearing new/unusual gear and living in a "Plan B" war tent. Thus, even if some of his gear were found near his body, or if he were found close to the remains of his tent or baggage train, people might have just not recognized the new stuff as his.
Anyway (or TLDR), as they had only the body to identify him, even weak evidence from his body must be written down