What is this part called on a bathtub handle
I'd call that an 'adjustable stop'. But it's not a standard part.  It's something unique to that exact model of faucet.

So you will need to find the faucet manufacturer and ask them for the replacement part as you're not going to find it in a hardware store.

If the faucets were low quality or just a very old model, I doubt the original manufacturer, even if you an find them, will have the part to just give to you.  You may have to buy a new unit of that model faucet just for it.

If you can't identify the manufacturer or thy can't give you the part, things get worse.

You could replace both faucets with new. But I know I wouldn't want to deal with the tile work.

You could 3d print a replacement part.  You'd have to carefully measure the existing one in extraordinary detail though, or find someone with a 3d scanner that can do internal negative space like that part has, which is  going to be tricky.

You could remove the part from the other faucet so both work the same way, which may be better than the current state.

You could try to find a small hose-clamp.  They have a similar shape, and the screw one ones can go on tight enough to use for something like this.

You could 'slice' the existing part into two, perpendicular to its axis and use one half on each faucet.  This may be tricky if the keying on the handle isn't full length.  If that's the case, you could wrap some electrical tape around the shaft or fashion a spacer to hold it in the correct position.

This last one is what I'd do.