You should be able to peen the rivet over and take up the side play. Lay it on a flat metal surface and use the ball end of a ball pein hammer. Use light strokes and check the play frequently. Stop when it's tight enough that the hooks can't overlap but are still free to move easily.
May be fine, does the shift drum have nice disc shaped sides that serve to keep these arms in place? That loose fit may be THE DESIGN to assure no binding. How did we come to have this apart? Symptoms? Just noticed it on the way by? SpringS OK?
It's supposed to be that way. There are other pieces that keep the arms where they're supposed to be when in use. Please don't go banging on it with a hammer. The shifty bits in a Suzuki GS500 (and many other bikes) look a lot like that, and basically there's a plate over the end of the shift drum that keeps the "hooks" at the end in the right place. The side-to-side play is of no consequence, and some play is needed to ensure the assembly moves freely under load. Also, you have omitted some VERY important information, like what sort of machine you're working on and whether you are having any other problems that would lead you to suspect this part.
pretty sure the 2 arms are on opposite sides of the shifter drum... they can never touch when assembled. they also prolly ride in a track to keep them in alignment & have springs so they follow the shifter cam on the drum
