For real astrophysical black holes, indeed the magnetic field associated with the black hole is really generated by matter outside the black hole in an accretion disk.
It is theoretically possible for a black hole in vacuum to have an electromagetic field. In the simplest case, for a non-rotating charged black hole in an asymptotically flat spacetime, one would have a Coloumb-like field around the black hole (the Reissner–Nordström black hole). This can be made more complicated if the black hole spins (the Kerr-Newman black hole) or if magnetic monopoles exist (a different version of the Reissner–Nordström black hole) or if there is a non-zero cosmological constant (the Kerr-Newmann--de Sitter black hole).
In this scenario (which is perfectly well-defined theoretically, but not realistic astrophysically) the key point is that the energy density associated with the electromagnetic field does not collapse into the black hole, but extends to infinity. The no-hair theorem tells us what properties a black hole can possess, and confirms that black holes can have electric charge.
Another way to express this, is that the electromagnetic forces between particles are mediated by virtual photons, not real photons. Virtual photons are not on-shell, meaning that they do not satisfy the relationship between energy and momentum in classical physics, and in particular virtual photons do not travel at the speed of light. In fact the speed of a virtual photon is not a well-defined concept -- it is not an observable quantity, and if you try to define a speed mathematically you will find it depends on conventions that you choose in doing the calculation, so cannot be a physical quantity. Rather, one can (theoretically) measure the electromagnetic field of a black hole by seeing how the black hole interacts with a charged particle -- a positively charged test particle will feel an electrostatic repulsion from a positively charged black hole.