I have addressed certain aspects of this in previous questions. Parts of this answer are adapted from this.

This is a great question. For my answer, I will focus on the 15th century, the height of the Milanese plate armouring industry. You hint a bit at the answer - mechanization played a part. Water-wheels powered the bellows of furnaces (in the 15th century, probably bloomeries, based on surviving armour) and powered trip-hammers that shaped blooms into sheets. Eliminating the need for bellows-tenders (while increasing power an allowing for continuous operation) and mechanizing the difficult process of flattening large blooms of steel greatly reduced the labor on the 'front end' of the manufacturing process and made raw materials cheap and more plentiful than they were before. This meant that while 6,000 armours (the total supplied by Milanese armourers, from stock, after the Battle of Maclodio in 1427) may take a lot of metal, but the mines an furnaces could produce the raw materials for that. Moreover, unlike some places (England, for instance, which needed to import high quality steel for armour from southern Austria) Milan was located very close to high-quality iron ore deposits in the Italian alps. This meant that a large bulk of high-quality steel could be easily transported to Milan.

On the other end of production, after armour was shaped, water-wheels powered polishing mills that mechanized the most labor intensive part of making armour - finishing the surface, which could be a majority of the raw labor that went into armour (polishing by hand is time-consuming).

Mechanizing the 'front' and the 'back' end of production left the middle, the work of the armourers themselves. This is where Milanese armourers really shone. Milan lacked stringent guild restrictions on subcontracting and workshop size (like Nuremberg did), and -also- did not have restrictions on makers of armour being armour merchants (like Cologne did). This meant that some armourer-merhants - particularly the Missaglia family - (I want to call them 'armourer-capitalists', really) could create a vertically integrate enterprise where they controlled furnaces, supplied their own metal to their own workshops or those of their subcontractors, and then finished it off in their own polishing mill. The subcontracting system is also important. Milanese armourers would hire workers as things like 'helmet maker' and 'gauntlet maker' and then a senior armourer, the 'traversator' would assemble the finished product. Contracts would be for a term (like a year) between one subordinate worker (or his entire workshop) and the employer. We also see evidence of this in the way that Milanese armour of the 15th century is marked - each worker would mark the piece they made, and then the master would mark each piece as a sign of quality at the end. In addition, many of the employment contracts survive, showing specialized workers making individual pieces. This specialized process meant that each individual piece of an armour might be worked on at once by specialists, which greatly speeded up production - it's not quite an assembly line, but it is a highly developed division of labor.

The total labor pool of armourers in Milan was quite large. There were hundreds of armourers working at any given time - in 1474 there were 72 masters, and each of these would have apprentices and journeyman (or other workers) under him. The labor organization outlined above allowed these large number of armourers to coordinate their efforts towards quickly turning out large amounts of armour. This made Milan the arsenal of 15th century Europe.

EDIT: I should point out that the mechanization of the manufacture of steel plates and of polishing is not unique to Milan. In fact it is seen in most (probably all, but direct evidence of water mills is sometimes lacking) of the major 15th century armouring centers. Cologne had polishing mills outside of the city. Nuremberg had both municipal hammer-mills for plates and a municipal polishing mill by around the turn of the 15th-16th century. Innbruck's armoury for the Dukes of Austria was set up with a polishing mill, as was Henry VIII's armoury at Greenwich. Water power for polishing and other tasks was a major factor in the siting of armouring centers.

Regarding labor, the division of labor in Milan is not entirely unique. Nuremberg with its tiny (6 people or less, as required by law) workshops was still able to produce large amounts of armour because different masters specialized in different pieces. However, in other centers restrictions on workshop size or on armourers trading in armour (as well as other political factors) prevent the kind of vertical integration you see in Milan or armourers gaining as much power as the Missaglia (or the Negroli after them) did.

These factors mean that while Milan produces probably -more- armour than other production centers and -definitely- exports it more widely (from England to the the remaining crusader statelets and eastern Venetian colonies), there are quite a few other armouring centers with truly massive production and a large export operation. At the end of the 15th century these include Cologne and Nuremberg in Germany, Brescia in Italy, and various centers in the low countries like Tournai (which add up to a decent sized industry, it seems). In addition there were local production centers in just about every sizable European nation and there was a successful 'boutique' armouring industry in Augsburg and Landshut, and the 16th century would see a large scale Ducal armoury established in Innsbruck and a 'boutique' armoury at Greenwich. Aside from these major players there were local armouring centers in many parts of Europe, particularly Germany. Armouring was a truly massive industry and while Milan was perhaps the biggest center in the 15th century, it wasn't the only one.

Sources:

Williams, Alan - The Knight and the Blast Furnace

Pfaffenbichler, Matthias - Armourers

Phyrr - Heroic Armour of the Italian Renaissance