This is an instance where we trade. Grammatical, clarity for linguistic convenience.

Properly speaking, we play music ON the piano. But for the sake of expediency, we just say, I play the piano.

Now you might ask, why don’t we say “I play the hockey stick? “ This is because hockey is a much more regulated game in terms of the instruments used to play it.

A better analogy would be to say, I play goalie, or I play center. Here you are stating the position that you play when playing the game. And so the instrument that am musician plays is more akin to the position they play when playing the game of music. But one would not confuse goalie for a game itself