Yes. Correct.
The digestive system of fish is just as strong, if not more, than that of birds.
Just like birds, fish ingest all sorts of things that would simply kill other animals. Stones, shell etc. They can digest just about anything. Most fish have a relatively small anal vent which demonstrates this. (for the kids watching this, the anal vent is the bum). Fish don't need a large anal vent because everything is digested and what is not used (the poo) is very small and easily passed.
Some fish, in particular most of the larger sharks, do not rely on digesting and passing alone. Although they can digest just about anything like any other fish, they have a relatively larger anal vent. This allows them to pass larger "waste" items. This is important because the sharks do not always digest everything they eat. Many sharks will expel (ie spew) their last meal if a new meal is there. However, if a new meal is not available then the last meal will have to last until the next new meal. Which means that passing half digested things might be needed.
Leaving a hook in a fish is not going to be an issue. Even if the hook is in the mouth and you just don't want to remove the hook is not an issue.
itsaboat mate...Life is just a boat and then ya marry one !