How come there is only one written musical language for western music, when there are so many different spoken languages?

OP, your question appears to be about musical notation (written language), not about the language of music per se.

Western musical notation developed out of Mensural notation , which in turn developed out of Neumatic notation.

The Byzantine Notation system is an independent development of neumatic notation, used in the Greek church, which looks nothing like Western notation. The Znamenny chant notation is the Russian derivative of Byzantine Notation. Hold that thought.

The comparison to make is not between musical written notation and different spoken languages. The comparison to make is between written notation of music and written notation of language.

The written notation of language is a script. In Europe, there’s Roman script, Greek script, and Cyrillic script.

There are many different spoken languages in Western Europe, but all use the same Roman script. They all use the same Roman script, because they all got literacy through the Catholic church, which used Latin.

The countries of Orthodox Eastern Europe did not use Roman script. They used Cyrillic (which was based on Greek), because they all got literacy through the Orthodox church.

There was an Ancient Greek musical notation, but it did not survive. Music notation in mediaeval Europe started in the church. And there was a Catholic tradition of notation (neumes, then mensural), and an Orthodox tradition of notation (Byzantine and znamenny).

Since all Catholic countries shared the same musical notation, it was straightforward for them to keep sharing the same musical notation, when it developed into something recognisably modern in the 1600s (even if by then music was also secular, and they weren’t all Catholic).

So there is only one written musical language for western music, for the exact same reason there is only one script for Western European languages.