you're right, it does look like it screws on.
the tap looks like a flashy stylish import, always a bad sign. You can trawl a few supplier websites to try to track down the maker.
Assuming it is a Eurean tap, the trhreaded stud will be metric. Metric threads are easy to match, you measure the external diameter and it will be 3mm, 4mm, 5mm, 6mm etc.
Metric nuts and bolts come in thread sizes called M3, M4, M5, M6 etc. You can probably guess what they fit.
Also test it with a magnet. If it is brass it will be rather soft and weak, and may have broken or worn.
You can then order a screw-on knob in the same size (I don't know where from, but you can try Lunn's)
A metric sleeve-nut might do, if the head is big enough for you to grap and pull or push. This type will screw well down the thread, so have a stronger fixing than an ordinary nut.
Example