"Renaming a schema does not update references to it inside function and procedure bodies written as string constants, since those are parsed only at execution time (see CREATE FUNCTION). Views and SQL-standard function bodies are updated automatically."
Schemas are not special in this regard. It's mostly incumbent upon the place that breaks this kind of expectation to note its deviating rather than expect one feature to be aware of the implementation details of another. In short, if the function/language pages don't cover that their string literal queries are not exposed for rename operations or dependency tracking that should be fixed in those locations.
