The short answer is yes (to which I'm tempted to add "of course!"). But it depends a little bit on what you mean when you say "genre of music." I'd bet that you're thinking about what happens when you go to buy recorded music: you can select from a bunch of options like "classical," "jazz," "easy listening," "world," "rock," and so on.

It's worth pausing for a second to reflect on what a weird set of technologies and social conditions enable that experience. To reflect on just how weird, historically, it is to think of music primarily as a commodity bought in recorded form and listened to privately on headphones. Until the 20th century, music didn't exist (couldn't exist) as something to be consumed from prepackaged recordings. Music, across history and cultures, is something that people did socially with each other--either as a communal activity or by hiring professional musicians to play live for a particular event.

And it's only in the present historical moment of being able to buy recordings of just about anything that the notion of "genre" really makes sense. (At least if I understand the intent of your question correctly.)

If you think about the nature of the different musical genres that you can buy, one obvious way in which they differ is simply when they're from. Classic rock is from around the 1960s, bebop is from a few decades earlier, "classical" is from roughly 1700-1900, and so on.

It shouldn't be too surprising, I think, that people have always been sensitive to the way that musical style changes over time. And the closer you are to a music's historical/cultural origin, the bigger distinctions you're likely to make. In the century of "Classical Music" most properly (the eighteenth century), musicians spoke of the stile antico (the old style) in contrast to the galanten Stil (the more modern style). (Here and here are examples of both taken from the same composition by GB Pergolesi, his extremely popular Stabat mater from 1736.) The stile antico traced its roots to the music of the 16th century, focused on strict control of dissonance and intricate intertwining of multiple melodies. (Such as "Adjuro vos filiae Jerusalem" by Palestrina.) The galant style, by contrast, was a fairly new musical development in the early eighteenth century which placed greater value on a single ornamented melody supported by a bustling rhythmic bass line.

The stile antico was more appropriate for sacred music, whereas the galant style was developed primarily for use on the theatrical stage. (Opera was a booming industry, eighteenth-century Europe's version of Hollywood.)

That brings me to the other, perhaps more important, way in which what we now call "classical music" was subdivided into many genres: its intended use. Because recording technology didn't exist, music wasn't just something you could put on in the background while you washed the dishes. You put in the effort to perform yourself or you paid someone to play music for a particular event. Music was composed for operas, for holding mass, for court dances, for private entertainment (that you'd play on your own virginal in your room), for convivial gatherings (you'd take out your part books & play a string quartet with your friends), and so on. Each particular use for music had its own expected instrumental forces (e.g. an entire symphony vs. an organ installed in a church vs. your own flute) and musical style. Music that's good for marching isn't so great for serenading your lover; and every dance step has its own characteristic rhythms. (You can't easily dance a minuet to the music for an allemande!)

So what we now call "classical music" is actually a huge collection of various genres (motet, dance suite, trio sonata, dramma per musica, symphony, fugue, divertimento, fantasia, string quartet, opera buffa, concerto, oratorio, and so on), all of which have different musical characteristics, intended social uses, and histories. Composers like Mozart took advantage of this, borrowing the stylistic characteristics of one genre into the music of another for the sake of expressive effect. (For example, this piano sonata starts off with a simple songlike tune but branches off into something "learned" and therefore vaguely stile antico before evoking the horn calls of a hunt and bursting into a quasi-dramatic "storm". Each one of these passages adopts a musical style reminiscent of a particular genre, so that the experience of listening to this sonata is a kaleidoscopic tour through many different sites of eighteenth-century musical life.)

This sorting of music into genres by their intended use is something that's still very much with us when we move music off our phones and back into shared social spaces. The music you'll dance to in the club is very different from the music you'll hear at worship or sitting around a campfire. But an ad on TV might adopt any of those styles as it tries to persuade you to associate its product with a particular range of connotations.