Is dark matter trapped near / in black holes, like regular matter is, or does something different happen with dark matter?
It interacts gravitationally.That's all there is to it.
There is a big difference though to the way that normal and dark matter interact with black holes - dark matter is far less likely to be captured.
Given two lumps of matter, one normal one dark, with the same mass and angular momentum with respect to the black hole; only the normal matter is capable of shedding its angular momentum (normally an accretion disc is involved), which allows its orbit to shrink enough to be captured (within three Schwarzschild radii). Dark matter is dissipationless; if it has too much angular momentum it won't get captured.