> I believe that definining x//y as math.floor(x/y) is also confusing
> in other cases (without being able to construct such cases right away).

In addition, defining x//y as math.floor(x / y) would break its connection with %:  a key invariant is that

    (x // y) * y + x % y should be (approximately in the case of floats) equal to y.

The connection between // and % is more fundamental than the connection between // and /, so in cases where the two disagree, the %-related one wins.

For applications:  it's true that they're not common, but they do exist.  One such is argument reduction:  e.g., for a toy case, suppose that you're implementing a function that computes and returns sin and cos.  The computation can be reduced to computing for angles between 0 and pi / 4:

def sincos(x):
    """ Compute and return sin(x) and cos(x). """
    q, r = divmod(x, pi / 4)
    <compute sincos(r)>
    <use symmetries and the last 3 bits of q to compute sincos(x)>

This is an example where if the relationship between % and // were broken, we'd get wrong results---not simply inaccurate, but completely wrong.

It's also worth noting that // and % are special in that they're the only basic arithmetic operations that can be computed *exactly*, with no numeric error, for a wide range of inputs:  e.g., if x and y are positive and x / y < 2**53, then both x // y and x % y return exact results.  Modifying them to return inexact results instead would be ... surprising.