My question flows out of the top answer to this question, from which I learned that a "linear function" is any function $f$ with properties of additivity and homogeneity of degree 1:
$$ f(x + y) = f(x) + f(y) \\ f(ax) = af(x) $$
Just reflecting on the formula for the median it seems to me that the function does not have additivity, but does have homogeneity of degree 1. Am I right about this? How can I prove what the right answer is?