I've never seen cabinet hardware fastened this way. You are correct in that it is like a thick brad, and as such it would be tightener the same way you'd tighten a brad. The problem here is tightening the prongs without marring the wood or the front of the pull. The factory probably used a special setting tool.

Can you use a "shim" under each rear prong, such as a small folded piece of matchbook cover to take away the play of the loose knob? It would remain invisible from the exterior. Very pretty piece.

What's the problem with it being "loose"? it's not going to fall off nor pull out. That's just how it was made.....I suggest you just accept it as it is....
However it appears to have been refinished....wondering if the hardware was also replaced at that time too?

Those are "Queen Anne-ish" drops on a piece of quartersawn oak in an Italian Renaissance style....?? Hmmm...

There's a faint circle around the drop in photo #2. I'm wondering if the drops are not the original hardware.

Thanks everyone for the replies. lindac92 I have decided to not worry about the hardware. At first it bothered me because the pulls are 2 pieces and the flower part rotated and I didn't want it to scratch it anymore. colleenoz, the faint circle around the drops are where the drops have been rubbing against the wood. Some of the markings are larger but those are the ones that are really loose. I have done some research on the drops and I could only find a company that sells them. They claim that they are from anywhere between 1850-1910. I would really like to know the age and style of the Buffet.
Here is a side view of the drawer