Two more "intriguing additions" for you, Robert:
In Britain, children ( typically siblings) also say 'Bags I the front seat!' or 'Bags I the last chocolate muffin!' for whatever you want to lay claim to. These expressions often vary according to region. At a guess, I grew up in a different region from Karen.
There's also the American expression - an extension of the one you heard - calling shotgun. "I call shotgun!" comes from the expression "riding shotgun", dating from the days of the Wild West when stagecoaches had an armed guard next to the driver.