You cannot feed your betta every other day. Unless he is bloated / having SBD, then you feed him every other day. Sorry, I speak in a blunt way. Can you eat only every other day? It wont kill you wont it? Same thing. They will be hungry, not enough nutrition. Fasting it a day gives the betta to clear all the unwanted food out. But right after the fasting day, you have to feed. Im not sure for the gold fish part, as I only keep bettas.
Pardon me for the crude way of speaking. Cheers!