Taking all the disclaimers into account that you're seeking medical advice on the internet and we could all be axe murderers for all you know;
No, not really. Humans can catch fish TB through open cuts, which is a real pain to treat, but it does require that it's in your tank to catch. Otherwise there's all the usual bacteria, but nothing drastically pathogenic in a fish tank.
It stings because it's less concentrated than you are. Humans are quite salty.
General advice to clean cuts stands, I personally use tap water to clean minor wounds.