There are some forms of both liquid and stretchy/self-stick electrical tape. The liquid is designed to replace the plastic outer shield but I doubt it will hold up long to the type of use shown in the picture. The other kind is often found in orange (for higher heat applications) or black for normal heat, such as your case. You simply stretch the "tape" (actually feels like a rubber and has no adhesive on it) around the surface to be protected and it will remain in place because it sticks to itself and will slightly adhere from the stretching action to the material beneath it. I have used it many times on aircraft application when I was in the military as an aviation electronics tech.
While either method may work, the problem is the stress you have placed on the wires. Wiring is never meant to bend at 90 degrees or even close to it, especially while flexing back and forth through regular use. That first couple inches of wire coming out of the adapter should be loose, not pulled tight. I would normally place a finger or thumb over the cord at about the 2 inch mark to hold it in place without straining it, then wrap the cord. That way the slack is not taken out of that first couple inches during the cord wrapping action.
Wires bent back and forth over time will all do that. I have fought with the wife for years about how to properly move and use power adapters for computers and how to use the machine while plugged in to avoid straining the connector or wires, or especially the connector on the computer end (non-Macs that have cheap plastic connectors).
