Yes, you can bring live fish and corals through TSA at the airport for US domestic flight. You can bring more than 3.1oz of liquid IF it has fish or coral in it (it means that the liquid is not dangerous and is life sustaining). Here is the TSA rule regarding live fish:
Different airline may have different rule towards bringing live fish onboard, from my reading, most will not have an issue as they have passed through TSA, but some airlines may charge extra for bringing a pet into the cabin. If you want to be sure that you are 100% compliant, it may be a good idea to call your airline ahead of time to see where they stand when it comes to having fish & corals on the flight.
This is ONLY for domestic flight within the USA. Obviously different rules apply to different country and also when you cross the US border.
This has been my experience talking to 4 different TSA agents (DC & Orlando) and 2 TSA managers (Orlando), and from my research online. As a disclaimer, I would say YMMV so please do additional research ahead of time.