I think music and other arts are important ways to express oneself like a language can do in every society.
In terms of social identity, I would like to mention three aspects of their importance; cultural identity, education and socialization.
First of all, since each society develops their own music and arts, there must be some difference between other cultures. Like my home country, Japan, has its own unique music and art. I remember when I visited the MOA museum in New York and it had a section showing only Japanese culture. I was so surprised how distinctive Japanese culture is because the culture is traditional to my home and I thought it was similar everywhere. The difference can lead to cultural identity. When I play Japanese traditional instruments, for example, it should be very unique to people from other cultures and make Japanese people nostalgic.
Music and other arts are strongly associated with education in a society. In Japan, many traditional songs are sung at school. This is how Japanese people share the same songs with each other even though their generation is different. When I provided a classical guitar workshop at a community centre, I used some Japanese traditional songs to play together. It is a party because everyone knows the songs and the melody lines are not so difficult. In that sense, education helps those arts to be shared all over the society.
The last one is socialization. In the first point, I argued that music and other arts can make cultural identity and that identity can be provided by education as the second point. Those shared music or arts can bring socialization. Even if people do not talk a lot, they can understand each other guessing that they share something. I believe that music and other arts play the role.