A confession: Since we started keeping fish in September, we have had at least ten fish die. Of the seven currently living in our aquarium, at least two look sickly. On the bright side, I've been watering my houseplants with the water we take out each week when we clean the aquarium, and most of them responded by putting out new growth well into the fall!
Unlike tap water, used aquarium water is almost ideal for plants. Tap water generally comes out icy cold and full of chlorine; aquarium water is generally treated to remove chemicals and heated to a comfortable temperature. The fish wastes in the water also provide a ready source of nitrogen and other nutrients to plants.
Beware of using aquarium water, however, if you add anything to it other than fish food or a water conditioner. Salt, medications, and, of course, algicides, can all be harmful to plants. You might also want to avoid pouring it on your lettuce right before you make a salad, unless you are prepared to wash it with soap.
Water full of fish wastes is not the same as the 'fish emulsion fertilizer' that you see advertised. That is made mostly from ground-up fish. These fertilizers are usually gentle and organic (with a NPK of 5-1-1 or 5-2-2), although I am not sure they would qualify as vegan. Aquarium water is also a very gentle fertilizer that can be used often, although container plants and houseplants will probably require an occasional dose of something more complete.
You could even try growing your plants right in your fish water, but although that apparently works quite well, it does get more complicated.