The quantity a = cJ/GM^2, the spin parameter of the Kerr metric, is not observer-dependent. No observer (or system of coordinates) can turn an axisymmetric system into a spherically symmetric one.
Also,
> Two black holes who differ only by their ... angular momentum ... are not distinguishable
is contradicted by
> Two black holes who differ in ... spin ... are distinguishable
(Spin) Angular momentum is J in my first line above.
Also,
> Two black holes who differ only by their ... angular momentum ... are not distinguishable
is contradicted by
> Two black holes who differ in ... spin ... are distinguishable
(Spin) Angular momentum is J in my first line above.