How can I tighten my shower head holder so that it doesn't slip and let the shower head drop out
From the last picture, it really looks like there used to be two "fingers" that engaged the ratchet mechanism.  One seems to be broken, which would also account for it no longer having the strength to hold the head in place.

If you're like me, you set this to the angle you like and never adjust it after that.  I would put some adhesive in the round channel outside of the ratchet ring and screw the parts back together, set the desired angle, and allow it to dry.

If you wanted it to be adjustable, you could fill that same channel with a small amount of rubbery adhesive (like a silicone sealant or shower caulking - something that doesn't get hard when it dries) and let it dry before putting the two sides together.  You just want a little rubbery adhesive in the groove to provide friction when the two pieces are screwed together.

Once it is dry, you can set it to the correct angle and tighten the screw to keep it in place.  If you needed to adjust it later, you might have to loosen the screw, but at least it's adjustable.