This website is the most detailed scientific study of the Jonah account I've seen: https://bookofjonah2amaic.wordpress.com/2013/07/03/is-the-jonah-and-the-whale-account-scientifically-and-historically-plausible/
Condensing the article, the author seems to believe that the sperm whale is the most likely candidate of whale for the biblical account (considering size and the locations of internal organs), so that's what he's mostly referencing, though your fictional fish can obviously differ.
There's a section that discusses both stomach acid and oxygen. From what I read, the stomach acid wouldn't be pleasant, but also not too deadly depending on the whale (your fictional fish could have a low pH acid for slow digestion).
Growing up in church, we had a youth pastor suggest that the stomach acid could have turned Jonah's skin white (which gave him a lot more attention by the time he reached Ninevah to warn them of judgment). A number of articles suggest this could have happened to Jonah (and did happen to modern men), but some of the articles are referencing hoaxes or fictional accounts--so I'm not sure where the "bleached skin" idea originally came from. Still, it's worth considering.
As for oxygen, the article I linked to above says this:
"Macloskie (1942) argued that the whale has to expel superfluous water from its mouth after receiving food. In the process a creature trapped in the mouth might reach the laryngeal pouch below the larynx. The pouch is big enough to hold a human who would, in addition, use the whale’s own air supply and have no worries about digestive juices.
He also suggests the possibility of living in the sinus cavities of the whale.
I guess in conclusion, your fish really just needs to have a low pH for the stomach acids and access to the whale's lungs or air. You could have the stomach and the laryngeal pouch connected (like the fish in the article). Having a dude in your lungs probably isn't super healthy for the fish, though, and he'd probably vomit your character out eventually, or die.
Also got to consider if his entire cabin is also swallowed, the guy has to survive being crushed by debris from his house. Unless your fish is a really...big...fish... But hey, Pinocchio's dad did it.