I think this is a 4 prong drive with an internal thread, so in order to get it off you will have to lock the spindle and then use a large spanner to loosen the grip. This will most likely take a bit of force. Once the 4 prong drive is off, you should be able to mount the chuck.
For small bowls on a lathe like this you will probably start out with the screw chuck shown in one photo. Drill a hole the size of the solid part of the screw in the centre of the blank, mount the screw chuck, then mount the blank onto the screw and bring up your tail stock (for safety), then shape the outside including forming a tenon to fit your chuck jaws. Then take the bowl off the screw chuck, mount into jaws and turn the inside. When finished, make yourself a large wooden disc that can be mounted either directly onto the drive spindle (using a suitable faceplate bought from a supplier or by providing either a tenon or a recess that can be held in your chuck jaws) and reverse the bowl. Best to cover this plate with some non-slip matting (from the pound shop), bring up the tail stock and then finish the foot of the bowl.
This is just one method, there are others, but it's most likely the easiest way to get to a finished bowl.