A bit late to answer I believe the origin of the piano metaphor is Notes From Underground by Dostoyevsky.

Even if man were nothing but a piano key, even if this were proved to him by natural science and mathematics, even then he would not become reasonable, but would purposely do something perverse out of sheer ingratitude, simply to have his own way…then, after all, perhaps only by his curse will he attain his object, that is, really convince himself that he is a man and not a piano key! If you say that all this, too, can be calculated and tabulated…then man would purposely go mad in order to be rid of reason and have this own way.

The broken piano key signifies the moment K stops acting in a way determined by his makers and becomes an agent of free will, kinda like pinocchio. You can see similar piano symbolism everywhere Westworld: They are building pianos.