Neither black holes nor neutron stars extend any kind of magnetic tendrils. There can be jets of matter that are ejected from the vicinity of a black hole (due to the Penrose process or something similar), and those jets can have electric charge, but that's not what your linked study is about.
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I have a (probably stupid) question - are the tunnels (https://www.indiatimes.com/technology/science-and-future/ear...) the solar system is currently in - connected to black hole at the heart of our galaxy?
Like do black holes extend enormous magnetic tendrils into the surrounding space similar to neutron stars?
Neither black holes nor neutron stars extend any kind of magnetic tendrils. There can be jets of matter that are ejected from the vicinity of a black hole (due to the Penrose process or something similar), and those jets can have electric charge, but that's not what your linked study is about.