What is the Relation Between Fat, Scar Tissue, and Collagen?

Fat is a living tissue made up of cells specialized to store unused calories as fat.
All injuries heal by deposition of scar tissue, a biological glue which is constantly modified over a 8 - 12 month period after the injury / operation. But as stretch marks demonstrate it is permanent. You cannot permanently erase scars or remove it.
Collagen is but a component of scar tissue. It is produced by and deposited by scar cells (Fibroblasts) found in nearly all tissues.
When we say that something "produces collagen" it is a clever marketing way of saying the product causes an injury resulting in scar deposition.
Obviously the kicker is HOW MUCH ? A little scar tissue UNDER (not in) the skin is not important and May even shrink the skin for a while, on the other hand a large injury (gunshot, scalpel, burn (Laser, Acid, or hot water) - ALL produce a LOT OF COLLAGEN. But, not exactly where we want it and how we want it.

Fat in the human body is referring to specialized cells that accumulate and store fat as energy for future use. It is fairly soft and is what is removed with liposuctioning.
Scar tissue is what the body makes to heal injuries. It does have collagen as part of it. Scar tissue will be found in any area of injury including liposuctioned areas.
Collagen is a protein that is found naturally in the body in many areas inclusing aroud fat and in scar tissue.