When I am running an Oracle Apex app in Debug Mode, then a black bar shows at the bottom of each page.
One of the links of that black bar is Session, where you can see the values of various session variables and other stuff. In that Session window, the last tab is errors -> that shows the errors encountered by each user for each page.
In which table are those error messages stored. They are shown in an interactive report - so I think that they are being captured in some Oracle table.