Michele Grosso and mmeent have relevant points. The thing that must be remembered is the event horizon does not in any way causally influence the outside world. It really is the other way around. There is then angular momentum associated with a Killing vector $K_\phi$ for the entire spacetime. This carries the angular momentum information with it, or in a Noether theorem sense defines an isometry that defines angular momentum as an invariant. This is what is relevant.
If we think of there being a Gaussian surface around the black hole that also acts as a sort of cloak, then it matters not whether this is a black hole or a compact object of the same mass and angular momentum. The source of the external gravity field is irrelevant. So it matters not whether there is a compact star or a black hole behind the Gaussian surface/cloak. If there is a black hole the we can think of the horizon as a congruence of null rays that spiral as a sort of barber pole. However, the causal influence is from material that went into the black hole, not the horizon influencing the outside world.
Ultimately a black hole is a quantum object, and its angular momentum is similar to the intrinsic spin of an elementary particle. As a result the angular momentum is ultimately not derived from anything, which Michele makes a correct point on. A complete understanding of a black hole according to quantum theory would most likely have the angular momentum as an eigenstate similar to intrinsic spin.