GLASGOW—Nothing says Scottish exceptionalism like a highly caffeinated, sugary drink that’s bright orange and tastes a bit like bubble gum.

Irn-Bru, the beverage in question, doesn’t have much iron in it, and it isn’t brewed. That’s why A.G. Barr PLC, which first marketed the drink as Iron Brew in 1901, changed the name after World War II because of proposed U.K. labeling regulations.

This...