Are a black hole's jets caused by the black hole's magnetic field?
You appear to have a thing about linking the magnetic field generated by Jupiter to that around a black hole (note that a field does not "belong" to an object) - the two things are entirely different, the only connection being that magnetic fields are generated in both cases by currents caused by the motion of charged particles.

In Jupiter's case, the magnetic field is (likely) generated via a dynamo process, involving currents that flow in its metallic hydrogen interior.

In the case of a black hole, as material travels inwards, it forms an accretion disk around the black hole (because angular momentum must be conserved), in which magnetic field can be generated and amplified. The twisting of the magnetic field lines around the rotation axis of the black hole could create a kind of magnetic funnel lined up with the rotation axis, along which charged particles can be accelerated by a strong magnetic field gradient. The field lines would not look like you have shown for Jupiter, they would be twisted into a helical topology along the rotation axis. This is part of the Blandford-Znajek effect, which I do not claim to properly understand.

So the answer to your question is maybe, but its very much a topic of contemporary research.