No, the hole you urinate from (the urethra) and the vagina are not the same hole. The cup will not get in the way, or fall out when you pee.
Some people prefer to remove their cup before a bowel movement as the pushing during a bowel movement may move the cup further down their vagina (but probably not enough to push it all the way out). If you do not wish to remove the cup but it has moved, you can adjust it by pushing it further back into the vagina after the bowel movement.