No, a classic GR black hole does not have its own magnetic field. It can have electric charge, along with mass and spin, but that's it.
There is a somewhat fringe alternative idea called a MECO, which can have a magnetic field. Unlike some of the other fringe black hole alternatives, this one at least produces falsifiable predictions.
I suppose what I meant to have asked was, can an external magnetic field intersect with a black hole's event horizon and then escape?
For instance, with these theoretical tiny black holes, what would happen if you held a powerful magnet close enough for the lines of flux to actually intersect the event horizon?
There is a somewhat fringe alternative idea called a MECO, which can have a magnetic field. Unlike some of the other fringe black hole alternatives, this one at least produces falsifiable predictions.