This is before mass media. A band would have no air play, no downloads, no CD sales or even vinyl sales. Beethoven was living at a time when copyrights were coming into more force, but royalties were not yet being paid. So, he made money by 1) being paid for playing concerts ( before he was deaf he was quite useful- not only played piano but viola) 2) composed for commissions ( arranging Scottish songs, for example) 3) composing for music publishers 4) donations by kings, aristocrats and 5) teaching students... which he didn't like.

So, if you liked Beethoven's music, you could take lessons from him ( which seems to have been hard to do) give him some money, go to a concert, sponsor a concert , or buy the sheet music and play it yourself. Playing it yourself would have been pretty common. Most people made their own music. It often involved others- there were musical societies, clubs, etc. that would get together and perform, and buy music...sometimes they were pretty good, sometimes they were not. And they liked to sound good regardless, so there was a market for what was called "gallant" music, stuff that was not too difficult and sounded good. Composers often did much better composing for small ensembles- Beethoven got more money out of his string quartets than his 9th Symphony..

Playing a composer's popular music might go beyond buying his/her published sheet music. If an opera had a very popular aria, or a glamorous new singer had captivated the hearts of, say, everyone at Covent Garden singing The Lass of Richmond Hill, that tune would get re-worked and arranged for piano solo,or for harp, or some guitarist might work up an arrangement, or it might pop up as a theme in someone's string quartet- or someone would just sing it walking to work, or over a beer in a pub. For example, right on the eve of the French Revolution, there was a light opera called Pierrot dans le Lune in France. The title song turns up in a fiddler's manuscript book in Flanders at the same time, arranged in two parts- ready for playing for a dance, or anywhere else.

Awesome question! Hope I'm not late to the party and that I also don't ruin it. This will be a long answer, but I will divide it to make up for a better reading.

"Band merch" started to be a thing with what we call in social sciences (predominately in visual culture and culture studies) as the "cult of personality".

A little history:
Before the French Revolution (FR), societal codes maintained and stratified society. For instance, people would be dressed according to their class and, conversely, there was attire you could only obtain if you indeed belonged to that specific class. So, somebody was dressed in nobility attire, you would recognise their status and know how to address them immediately. Social interaction were also kept segregated, so it was not common for the working class to attend a concert or know the music of, say, Beethoven. That was a pleasure only people of a specific category would have.
After the FR, these class norms are dismantled to give way to the rise of bourgeois society: clothing starts to be mass-produced to "standard" sizes (much like today); seasonal wear intensifies; social interaction no longer relies on your social statues. This all accounted for you not being able tell what was your neighbours' place in the society, so "personality" comes into question because, ultimately, you depended on it to standout and separate yourself from the others—if you wanted that kind of attention.

The first musician to ever reach "cult following" was Paganini. He was a virtuoso in all right but bourgeois people who attended his shows were usually not versed in music. They were looking for a show and Paganini's playing was as much of a spectacle as his music. For classical musicians, this was not posible as things like posture, composure and societal distance influenced their behaviours widely. It was also not common for musicians of a certain fame to play their own music. They would compose, direct, arrange and produce their shows but they would not play it live. Paganini would be there in person and use the "variations" to convey emotions. Obviously, a lot more people had access to Paganini's shows than Beethoven's (in his lifetime), so his success was not exclusively a matter of his talent as a composer but had a lot to do with his distinctive personality—or, so to say, his self-marketing skills.

The "cult of personality" increased hugely with mid-18th century intellectualism and with the standardization of education—the first official "public schools" are from this time. It's during this time when national heroes are "canonized" according to both nationalistic ideals and the interest in "historical subjects". The idea of a "genius artist" usually comes from this time: some of the great artists of the past had been either forgotten or were only consumed by a special few and it was the study of their work that motivated their rise to that status.

Modernism impacted this idea hugely, with the colapse traditional artist schools. Up until this moment, smaller artists would cluster around a bigger figure who had mastered the skills of the Grandmasters. This person would eventually teach and form a "school". Some of these smaller artists would never rise to fame and some would even spend a lifetime working for other bigger artists. But Modernism offers the possibility of every artist (small of big) to have a participative role in art creation, as it motivates artists to look for their specific art research or approach, so the idea of a group of artists working towards the maintenance of a classical style eventually dies out. Monet, for instance, was generally despised by the Parisian society, but loved by Baudelaire for committing to his own approach. This is specially true for late 1800s and early 1900s "-isms" art movements. Picasso is usually looked at as the "superstar" of Modern Art because, even though we worked alongside many artists, he joined and abandoned art researches whenever he felt like he had accomplished everything we wished and he developed a singular style every time.

The direct support of an artist's career by the general public was not common until the 1950s. Picasso, Dali and Warhol were specially famous for this as they used affordable techniques to mass-produce artworks for "general" consumption or also engaged in marketing campaigns once they were established, at the while offering "premium" artworks for galleries and patrons. For music, the equivalent to this has to do with the commercialization of records.

Up until this moment, you could only "publicly declare your support for an artist's" work if you invited people over. At the same time, music only started to have an influence in opinion-making in the late 50s and early 60s, then again, with the "cult of personality" and phenomena like "the beatlemania". "Band merch", on the other hand, was presented as a sort of statement, not so much as a means to show your appreciation for a band, but rather because it identified you with a "movement". So, in this sense, you were the one trying to say "I'm special". Mostly, it has to do with the subculture of heavier genres taking a different approach to the traditional structure of pop music: music that was politically-engaged, violent, unpolished, philosophical. This drastically increased after the "demonizing" effect the media had on such bands but it started as a bootleg/DIY punk idea. Some artists in the 80s also saw this as a possibility for making an extra income to make ends meet with all the touring and record label expenses while consuming large quantities of drugs. This is when you start seeing the rise of "souvenir" merchandize—that t-shirt with the band's logo on the front and the tour dates and places on the back—to provide the consumer a way to communicate the general public "I'm not only dedicated to this band but I have also been to their shows".