It is just the way we say things. "She plays violin." = "She plays the violin." = "The instrument she plays is a violin." It means that she plays an instrument and the instrument is a violin and she plays it in general. It could be at any time. "She plays a violin." could also be used to mean she often plays violin. However, it is used more to refer to a particular place and time. "She plays a violin in the orchestra." = "She plays violin in the orchestra." You also might be referring to a particular time. "Last night, she played a violin." Note that there is no hard rule. We could have said "Last night, she played the violin." - without talking about a particular violin. In slang "She plays a mean violin" = "She plays the violin very well." Hope this isn't too confusing.

This is a stock phrase - to play the violin, to play the euphonium. If we use 'a', the violin could be any violin (indefinite).