A Fire-King mug from the mid to late 1970s with a Duncan Yo-Yo Tourney Contestant imprint in red. Fire-King mugs were made with low-expansion borosilicate glass, perfect for hot drinks.
The model in this exhibit is a square-handled "Standard" coffee mug, 88 mm tall and 84 mm diameter. The mug holds nine fluid ounces. A Standard mug has a flat bottom inside and is sometimes called a shaving mug.
As to the "why" of this item, it is unknown. While the mug may have served as a gift for a contestant competing in a Duncan yo-yo tournament, it is more likely that the mug served simply as a marketing piece for Duncan.