I guess tha the "coaxial" part you refer to is actually just a shield. The shield is made from a fine wire that is "braided", and runs the length of the cable. I'm not privy to Apple's designs or schematics, but most likely the shield is there for EMC (electromagnetic compatibility), and not as a current-conducting ground wire. Using a shield as a conductor would be unusual to say the least.
That said, any damage to the shield is unlikely to affect the function of your cable.
However, the insulation on the wires in the cable should be checked for evidence of cracking, fraying or other deterioration. It's important all of the inner wires (inside the shield) are in good condition. You have done the right thing by mending the cable with electrical tape: it's an insulator and it's flexible. As long as the electrical tape is in place, and in good condition, I don't believe the cable poses any risk to you, or to your Mac.
I had the same problem as you recently on my Magsafe cable. The outer insulation in one area on the cable just began breaking down, and falling off the cable! I've never seen wiring insulation behave like this. I also used electrical tape as a temporary solution. I was able to find this replacement Magsafe cable on Amazon. This cable allows me to use one of the new USB-C chargers to charge my 2011 Macbook Pro. I've been using it for several months now & it works.
It appears from the picture you have the newer Magsafe 2 connector.. Consequently, the cable I bought will not work for you. However, it seems they also have a version for the USB-C to Magsafe 2 cable which may suit your situation.