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The French word that was translated to stud was ferret. I don't actually see any justification for translating ferret here as stud, as I will explain later.

To see what ferret means, let us look at the definition from the Larousse dictionnaire de français. This is

Petit bout (en métal, en pierre précieuse, etc.) d'une aiguillette.
Small tip (of metal, gemstone, etc.) of an aiguillette.

So what is an aiguillette? The English translation is aiguillette, but this won't help modern readers because the word has largely fallen out of use in both English and French (because aiguillettes aren't used much anymore). Larousse says that an aiguillette is

Cordon, ganse ou tresse, ferré aux deux bouts, ayant, jusqu'au règne de Louis XIV, servi à fermer ou à garnir les vêtements.
Cord or braid, tipped with metal at both ends, that until the reign of Louis XIV served to fasten or decorate clothing.

So an aguillette was a small length of cord or braid, often tipped with precious metal or gemstones at both ends, that was used to fasten clothing closed, or which was worn purely for decoration.

One can see pictures of them in English Wikipedia. This says

Portraits of the 16th and 17th centuries show that aiglets or metal tips could be functional or purely decorative, though many were used to "close" seams and slashes that are not always apparent on dark garments in portraits. They were made in matched sets, might be of silver, silver-gilt, or gold, and were worn in masses.

Here is a 1562 picture of Lord High Admiral Clinton wearing a doublet fastened by aiguillettes, from Wikipedia.

Lord High Admiral Clinton in an arming doublet of 1562

So were the diamonds on aiguillettes? If they were, stud might be a reasonable translation of ferret, because a stud is a piece of jewelry that you use to fasten two buttonholes together, and similarly an aiguillette is a piece of jewelry you use to fasten your clothing.

However, in this case, the ferrets were not on aiguillettes, although they were indeed on the ends of ribbons. A later description of the diamonds in Le Comte de Monte Cristo follows:

Tenez, lui dit-il en tirant du coffre un gros noeud de ruban bleu tout étincelant de diamants; tenez, voici ces précieux ferrets ...
“Here,” said he, drawing from the casket a large knot of blue ribbon all glittering with diamonds, “here are the precious ferrets ...

Note that the French word noeud is broader than the English word knot; a bow would also be called a noeud.

So the diamonds are in a large decoration made of blue ribbon. We also find that this was meant to be worn on a shoulder. This decoration must then be made of six blue ribbons knotted or woven together somehow, and the diamonds are on pieces of jewelry which are affixed to the twelve ends of these six ribbons. They may be intended to dangle around the shoulder like epaulettes.