Sharks normally respond very quickly to the scent of blood in up to a mile radius, but what if the shark itself simply got injured? Would it respond to its own blood at all?
I'm pretty sure, they can smell their own blood, but they do not attack themselves. Wounding a fish results in the release of chemicals into the water. This is a signal for other fish about potential danger and they will escape (Schreckstoff). Of course, the predator of the wounded fish will not mind. So this seems to work best within the same species. Fishermen and military have apparently used dead or rotten sharks in the past to keep sharks from attacking (Shark repellent). I guess, a similar mechanism plays a role here. Sensing a wounded member of the same or close species (including oneself), would block any feeding behavior.
The Myth Busters have actually done this (not very scientific) experiment: https://youtu.be/E1GenfKR9pk
It is likely not the case that there would be a feeding frenzy if the question is whether a shark would attack an injured member of it's own family or species. The main reason is for reproductive means as a video will explain on this website: https://sharkopedia.discovery.com/shark-topics/life-cycle/#how-do-sharks-avoid-inbreeding
Although there could be a few exceptions. Semelparity in female tiger sharks for instance, has been observed. Information about all sharks pertaining to this matter is difficult to come by.