How do you repair damage in a laminate floor since the decoration is actually very thin?
The putty solution
You could gouge out the problem and apply a hard putty. Plastic Wood won't stick well enough but you could use some non-shrinking wood fillers which are not quite as strong but take stain better, or epoxy putty that will be plenty strong enough but won't take stain well.
The real task here is not a woodworking one, but an artistic one. With an artistic worker willing to recreate the colour and grain pattern of the surrounding flooring, you can stain or paint over the filler. Then cover that with a couple of coats of polyurethane floor finish to protect the colour. This is not easy to get the colour and pattern right, but it can work. Look really closely at the patch in the fourth photo above, half is just the base stain, but the other half has almost dissapeared.
The inlay solution
If you are more of a woodworker, you can use a router to do an inlay job.
First make a "U" or square "donut" shaped frame to control the router. Cut out a square that follows the board patterns to about half the thickness of the flooring. Square out the corners with a chisel.
Then cut right through the flooring for most of the centre of the patch, leaving a shoulder all around.
Then cut an extra piece of the flooring to fit perfectly into the patch. You will have to cut a shoulder under this piece so that it will fit into the inlay cut and overlap on the shoulder for a gluing surface.
What is nice about this technique is that you don't have to find a piece of wood that matches as long as you line up your cut-out with a full board or two pattern in the original floor because the new insert will simply look like another short plank in the floor.
Use waterproof glue and a weight to set it in place and the patch will be invisible, although requiring some careful craftsmanship.
The replacement solution
If you are using the new clip lock type of laminate flooring that are not glued down nor glued together, you can simply disassemble the floor up to the damage, replace one single board and put the rest back together. Now there's an interesting advantage for the clip lock system and a reason to keep a spare piece or two stored away.