I tested this with P2003 and didn't see what you're seeing.
I created some tasks with a variety of links (FF, FS, SF, SS). Then I highlighted the entire row for each of the tasks I wanted to copy (five rows in all), pressed Ctrl+C, switched to the other project, clicked on a blank row, pressed Ctrl+V and all tasks and all links were copied and created correctly. The ID numbers were correctly changed to reflect the new location that the rows were in the other schedule.