What follows here is a very long (for reddit) but nonetheless certainly incomplete examination of the city's history. To talk about modern industries like fashion, we only need to examine the industrial era; however I will include a follow-up with more general Milanese history and a brief examination of its role in Italian culture and society below.
Milan has always had a special, and very prominent, place in Italian history, especially in the realm of business and culture; influential and high-profile industries like fashion are no different. At the moment of the very foundation of modern Italy, even though the new Kingdom of Italy had set its sights on Rome capital from the start, Milan was awarded the enduring nickname of “Moral Capital” (in Italian, Capitale Morale). As in, the city had lost the "competition" to become capital, but was nonetheless the "Moral Victor." Indeed, Milanese entrepreneurs, newspaper editors, artists and writers were perceived as important economic and social leaders. Milan’s network of canals, technical and vocational training institutes, and its sheer size made it an ideal hub for the new country’s industry. Its position as the natural starting point for rail and telegraph lines into Germany and France also made it the natural home for foreign banks and investors. At the turn of the century, contemporaries like the writer and Sicilian transplant Giovanni Verga affirmed it was “The most city-like city in Italy!” When in 1906 the Kingdom of Italy wanted to show the world what progress it had made as a unified country, authorities decided to stage the World’s Fair, and there was no discussion that Milan was the ideal host city.
Milan’s industrialization was accelerated by the needs of the first world war, consecrating Italy’s “industrial triangle” marked by the cities of Milan, Turin, and Genoa. In the Second World War, Milan’s industrial importance and its unexpected role as capital of a fascist redoubt made it a prime target for allied bombing. However, after the war Italy's rapid integration into the global economy created a prolonged economic boom, and Milan was at the heart of the social and economic changes which turned the country into a leading modern economy. People from all over Italy migrated to the city looking for work, and a popular saying soon emerged commenting how hard it was to find a native Milanese person in Milan. By the 1980s, the city’s factories had moved to the suburbs, and advertising agencies, television networks, as well as fashion houses conspired to develop a glamorous image for the city. In the '80s and '90s, the city developed a voracious consumer culture and with it a vibrant nightlife. In the 80s, a commercial for the very Milanese Ramazzotti liqueur described the city as “Milano da Bere,” a phrase meaning “Milan to Drink” (you can watch the commercial in question here) The phrase would quickly become shorthand for the city's glamorous nightlife-driven culture. In brief, Milan consecrated itself as a sort of aspirational capital of capitalism; Italy’s window on world; a sort of Italian "New York."
So when an entrepreneurial man running a leather business in Florence decided to make his bid to breakout into the world of high fashion, he would need bankers to finance his business, models to wear his clothes, and an airport nearby so he could meet international buyers. All these things were present in Milan, and it's no surprise that man in the tannery business from Florence with dreams to build a global brand did move to Milan: his name was Guccio Gucci. The same logic drove Gianni Versace to Milan, in addition to the sons of Ermenegildo Zegna, and countless other entrepreneurs, designers, and stylists. Many Italian fashion houses, like Gucci, nonetheless kept a presence in their hometowns; this is fairly common in Italian business, especially luxury goods, where ties to history and heritage are very important. However, the fashion industry is a business like any other: and in Italy, most business happens in Milan.
However, while Italy has always been an exporter of luxury goods and Milan had always been Italy's industrial epicenter, until the 1980s the Milanese fashion industry was much less internationally recognized than that in Paris or London. In fact, some Italian "high fashion" brands trace their origins to importers of foreign fashions (like the aforementioned Guccio Gucci). The push to internationalize the domestic fashion industry was part of a broad policy conceived at the end of the 1970s to revitalize the Italian economy after the boom of the 1950s and 1960s came to an abrupt halt and serious international shocks like the 1973 Oil Crisis had dampened prospects for future growth.
Over the course of two decades, Italy's large industrial conglomerates (many with the state as a major shareholder) had gone from being industrial powerhouses to being over-bloated and unable to adapt quickly enough to the needs of an increasingly complex global economy. In an attempt to distance themselves from the lethargic state holding company in Rome, a largely ground-up initiative was conceived by a number of city-level administrators in order to boost the visibility of entrepreneurs outside of Rome and kickstart export-driven growth. Foremost of these cities was Milan, where efforts were spearheaded by mayor Carlo Tognoli, who would later go on to serve as special Minister for Urban Policies in the De Mita government from 1987 to 1989. The idea was to organize conferences and events, ideally exploiting Italy's cultural patrimony in suggestive event spaces that would showcase Italian firms with high potential for growth. A new interest in the European integration process pushing through export-friendly legislation also helped generate interest, both on behalf of Italian firms and foreign buyers. While not all industries were able to fully exploit these initiatives (and as time went on participation in these events became a backchannel for firms to funnel dark money to political parties, notably at the Socialist Party national congress of 1987) given the fashion industry's focus in image and presentation fashion houses were some of the best poised to exploit the initiative. While multiple cities were able to raise the profile of existing conferences and trade shows, like the Pitti Immagine series of conferences held in the Pitti Palace in Florence, the largest beneficiary was the Fiera di Milano, a large conference center in Milan. Narratives about artisanal heritage, locally-sources quality materials, and designs informed by the country's rich artistic history quickly captivated the attention of buyers from the world's premier retail chains. Soon, even firms that only produced intermediate goods, like leather or wool, were renting offices and showrooms in Milan, and hiring sales reps and designers to manage own fashion brands in order to get in on the action (fashion is, after all, a high value-added industry with enormous margins).
The fashion industry had previously been characterized by small retailers and local tailors (the few department stores active in Italy mostly carried upmarket foreign goods) but in the 80s, entrepreneurs moving to Milan found everything they needed to turn their tailoring shops into global fashion houses. Milan's bourgeoisie of white-collar workers and professionals were already Italy's most prosperous consumers, and acted as willing test-samples for new product lines. Services like banks, law firms, consultancies, and most importantly recently deregulated television studios and advertising firms allowed for synergies that no other city in Italy could offer. Likewise, the city's status as Italy's "Window on the World" offered a trampoline to international markets; it is easier open up a showroom in New York with a loan from an American bank if you already have a credit line open with their Italian branch in Milan (and why not poach an American expat at the branch to be CFO of your new US subsidiary while you're at it!).
In my closing thoughts, I also want to mention a more recent phenomenon that has alimented the growth of the Italian consumer culture, and naturally the fashion industry: In the late 80s and early 90s there was an unprecedented rapid expansion of the Finivest group. Finivest was initially a real estate company that specialized in gated communities for high-income buyers, but in the 80s began expanding to own television networks, an advertising agency, print media, a football club, and even supermarkets. The Finivest group achieved a unique centralized ability to coordinate media influencing the social narrative of consumption, to an extent unparalleled in any other western economy. And all this was occurring just as the state-driven component of the Italian economy was being sold off in a series of privatizations. Thus the traditional Italian attention to craftsmanship was equated with attention to brands and logos, and the cultural propensity for "cutting a good figure" (fare bella figura) was equated with wearing expensive clothes. And nowhere was this more evident than in the city that Finivest called home, you guessed it: Milan! Paul Ginsborg, probably the most prominent historian of contemporary Italy, examines this social phenomenon in his biography of Finivest group's founder, Silvio Berlusconi, in the eponymous book published in 2004 (Ginsborg's Berlusconi, which I can't recommend enough, also offers a succinct introduction to Milan's role in the Italian economy).