The same criteria for choosing between whichever and whatever apply here as for choosing between which and what. This is a question that has been asked several times on ELU, including :
"Which" vs. "what" — what's the difference and when should you use one or the other?
In summary, which is generally to be preferred when choosing from among a restricted set of options known to both interlocutors. So in the nursery rhyme about the aggressive fish, it is:
Which finger did it bite? and not What finger did it bite?
In the exchange about music, the answerer is not restricting the choice of music to a limited number of options, so whatever is the natural choice.
It is possible, however, to conceive of a context in which whichever fits:
Is it OK if we listen to Blood on the Tracks or the new Van Morrison?
Sure. I'll listen to whichever you'd prefer.