The special root is called a rhizome.

You can separate them without hurting them. Ideally such division would be done after growth stops (during dormancy) or after flowering. You want to just water well, then lift and pry the root ball apart into separate plants and trim any damaged/dead roots from the divisions, and repot.

The roots in the picture seem to be simply fleshy, thick, tuberous roots. The special horizontal root a spider plant also has is called a rhizome.

You can separate them without hurting them. Ideally such division would be done after growth stops (during dormancy) or after flowering. You want to just water well, then lift and pry the root ball apart into separate plants and trim any damaged/dead roots from the divisions, and repot.