[pdonis]

> in what sense is a black hole "spinning"?

There are several equivalent ways of stating it:

(1) The hole's spacetime geometry is axisymmetric but not spherically symmetric;

(2) The hole has a Killing vector field that is timelike at infinity but is not hypersurface orthogonal;

(3) The hole has nonzero angular momentum as viewed in the asymptotically flat region at infinity.

> Can't we just pick as a reference the black hole, and say that the everything else is spinning, just as well?

No. All three of the above conditions are invariant and only depend on the spacetime geometry; they are unaffected by any choice of coordinates or the motion of any outside observers.