We just refer to music made before the 20th century as 'classical'

Not exactly, no. Classical music is "art music" linked to the "western tradition(s)." Not all music from before the 20th is considered part of that set, and a hell of a lot of music from the 20th and 21st centuries is classical music.

did they also have their own genres?

Yes, kind of. Genres are difficult to define, are pretty arbitrary, and change quite often. Also, today most people use "genres" to refer to music AND subcultures.

Genres are a way to try to classify music, and classifying anything is a difficult. Classifications are often arbitrary and controversial, they overlap and contradict themselves, there many ways to make them...

Was all music in the past the same? Certainly not. There were different types of dancing music, different types of religious music, different types of piano music, different types of orchestral music and so on. The type of text, the social setting in which the music was performed/used, the rhythm, the instruments, the musical logic in the work... A lot of things were and are used to classify classical music both of the past and the present.

Do you think some of our music could be 'classical' in another 200 years?

Well, we can't see the future but we can can think of a few things.

Our idea of "classical music" is complicated. The elites have had their own things to differentiate themselves from other people, this kind of music was the music of the rich. So who knows, maybe things will change so much in that regard that they make a new category for this music.

Before the 19th century, people were not particularly worried about this "classical" music idea. Music was music and that was that. Very little music from the past was preserved, and people were interested in presenting new music. They were not organizing concerts to present music from 300 years ago. People tried, and kind of failed, to keep religious music from the past "in its pure form." What if we forget this insane idea of keeping old music and focus on new music?

Classical music and other types of music shared a lot of elements in the past. Even if there were very noticeable differences, classical music really took a different path in the late 19th century. Just compare this music from 1925 and this "classical" music from the same year. New classical music has not really made it to the public. It's as if it had frozen in time in the 19th century (like your question mentioned, there seems to not be any new classical music). So, if it only stays alive in music schools, it might become an endangered species. I don't know if we would have classical music (from before the 20th century) today without its academic study.

On the other hand, today we have massive archives of classical music available for free. Seriously, musicians and musicologists of past decades would have killed for something like the Petrucci Music Library or Youtube channels. You have a hell of a lot of kids playing classical music (mostly old one), some like it and keep listening to some of it in their adult lives even if they quit playing decades ago. You have kids looking for that tite as fuck tune they heard in a video game, anime, movie or whatever, they like it and look for more... You have programmes aimed at disadvantaged kids in some countries (if those do any good is debatable)... I think classical music has become very accessible in the last decades, and it has the biggest audience in history.

So, who knows what will happen...

If you ask if we are adding new items to the "classical catalog," yes, we certainly are. Will that music stay in the "catalog" in 200 years? Maybe... See, some famous classical composers from the past are pretty much forgotten today. Their lives and works are mostly discussed by some music scholars, but other than that nobody cares about them. J.S. Bach was not really famous during his live, and his works faded away (it was mostly music geeks who kept them alive), and today he is a music god whose music is studied in pretty much every music school in this planet (it doesn't matter what instrument you play, or what specialty you study, there's Bach in your life as a classical musician, and even people in other musical traditions know his music).

I don't know if "ice cream car" famous is something anybody can aspire to.