Is this a stabilizer link?

Yep, that Technicolor purple item is the link. From the look of that picture yours are shot. If you're replacing those end links you should replace the mount bushings as well. They are the ones with the steel brackets that have rubber bushings around the bar and bolted to the frame.

Easy to change. New kits will come with the rubber parts, through bolts and bushings. I don't fight with the old ones, just grab the cut off tool and a couple quick cuts and the old junk is off. Then just put the parts on in the correct order and tighten the nut to spec.