It depends on where you live.
In many countries regulations determine when you are allowed to label a product as 'organic', but there are countries where the term organic is not subject to any laws. When organic is a protected label, this usually means that one is not allowed to use synthetic substances for example, but there are always exceptions and these exceptions also differ per country.
For the European Union the following applies:
EU rules do not allow for plants grown hydroponically to be marketed as organic. This is because organic production is only possible when plants are grown naturally in soil. This regulation also applies to plants that are grown in an aquaponics system.
However, fish which are grown in an aquaponics system can be sold as organic if the relevant legislation for organic aquaculture is followed. (source)
However, for the US there is no such regulation and hydroponically-grown foods can be labeled as organic if all US organic regulations are followed. There is however a debate on this so it's possible that this will change.
AFAIK 'sustainable' is not a regulated label in any country.