That's a rubber flex collet tap chuck. There should be a two-part piece in the cross hole that can be adjusted to grip on the square of the tap shank. You can get different collets for various sizes. It would live on a drill press with some provision for reversing for power tapping. It will drive a tap positively vs a drill chuck that can slip. Slip can be desirable, as that kind of chuck will snap a tap right off in a blind hole if the operator makes an error.
