As other answers have explained, surviving within a whale or a fish is not plausible without either a new kind of fish or some expensive equipment. Since this person is presumably not a professional fishgut-diver, we need to focus on the qualities of a fictional fish that a human could plausibly survive inside.
He is not inside the stomach, but rather the swim bladder.
Many fish possess a swim bladder - a gas-filled organ that helps them balance out their weight and remain buoyant, making it easier for them to swim. Some fish - called phytosomes - have a direct connection between the stomach and the swim bladder, not entirely unlike how land animals have a connection between the stomach and the lungs. These fish are able to fill their swim bladders with air by swimming to the surface and "gulping" the air. Our fish explorer was not "swallowed" so much as inhaled, which is how they end up in the swim bladder instead of the less habitable stomach.
The details of gas in the swim bladder vary from one fish to the next. Some use their swim bladders for oxygen exchange, others will actually fill it with oxygen from their blood, though they don't usually keep it oxygenated to any particular level, it just happens to be the easiest gas they have available (your person may actually have a problem from too much oxygen). Some will store oxygen in it when they are in shallow water and then consume that oxygen later when they dive deeper. Many don't care about the particular gases in it at all since it's just ballast to them, they just fill it up from the atmosphere and leave it alone - this is probably your best bet for survival, since although the air is limited it is at least still atmospheric air.
Which particular kind of fish you are dealing with depends on how long you want the person to be able to survive, but it is (somewhat) plausible to come up with a giant fish that a person could live in for a fair amount of time.
If you want to get really interesting maybe this fish has an air-breathing symbiote living in its swim bladder; this would explain why the fish maintains a breathable atmosphere inside and also give your person something to interact with (or worry about).
