The first thing to recognize is that genres are not scientific and are not necessarily even technical. A genre might collect songs that have a similar sound solely because they were all made in a similar time and geographic location.
So the short answer to your question is: if you can find songs tagged ”Genre: Soundtrack” then yes, Soundtrack is really a genre.
A music genre is a conventional category that identifies some pieces
of music as belonging to a shared tradition or set of conventions.
The shared tradition and set of conventions of the music in the Soundtrack genre is that the songs were all made to be the soundtrack of some visual media. That affects every aspect of the construction of those songs. For example, they tend to have muted lead parts compared to other genres because they don’t expect to have 100% of your attention. They tend to ebb and flow in a way that follows the ebb and flow of storytelling. They tend to have elements that build excitement or release tension along with the visual media they were written to accompany.
genre should be defined as pieces of music that share a certain style
or "basic musical language."
This is entirely consistent with the previous quote. Muting lead parts so as not to interfere with spoken dialog is part of the “basic musical language” of the Soundtrack genre. Using contrasting soft and loud parts, using a lot of textured strings and other pads, and builds and releases are part of the “basic musical language” of the Soundtrack genre.
Maybe you are conflating “musical language” with “music theory.” Not the same at all. Musical language is much broader and includes elements of the performance and production.
For example, in the Dance genre, the music was all made for dancing. A danceable beat is part of the musical language of the Dance genre. But 2 arbitrary dance songs aren’t necessarily similar from a music theory perspective. One might have a strong quarter note feel and be written in a major scale, while another might have a triplet feel and be written in Egyptian scale, and both might use entirely different instrumentation. You could give the sheet music — which typically does not even include the drum parts — for both songs to a musical genius and they might not be able to tell you that both songs belong to the Dance genre. Music theory is just paper. It is less important than the performed music, not more important. The musical language that defines a genre includes aspects that don’t exist in music theory. For example, the Grunge genre refers in part to using a particular kind of guitar sound that was popular in Seattle in the 1990’s. Guitar sounds are not part of music theory, they are part of performance and production.
One way you can see that genre does not matter to music theory is that you can give the sheet music for a particular song to a band made up of skilled musicians and say “play this in Jazz style” and they can play you a Jazz version of that song. Then you can say “play it in Bluegrass style” or “Disco style” and so on and they can play the song again for you from the same sheet music in those different genre styles.
Further, most genres are named by people who are completely ignorant of music theory. Often they are listeners and DJ’s and music critics.
So you have to take them with a grain of salt. They are informative, not scientific. But you can make a playlist of all Soundtrack or all Jazz and it is markedly different than a playlist of just random arbitrary songs.