The difference, by the numbers
Both have simple syrup. Only the Gin-Gin Mule has lime juice, mint, gin and ginger beer; only the Gunshop Fizz has Peychaud’s bitters, lemon juice, strawberries, cucumber, citrus peel and Sanbitter. In parts, a Gin-Gin Mule is ¾ lime, 1 syrup, 1¾ gin and 1 ginger beer; a Gunshop Fizz is 2 Peychaud’s bitters, 1 lemon, 1 syrup and 2 Sanbitter. By volume, the Gin-Gin Mule is 39% spirit, 22% syrup and sugar, 22% mixer and juice and 17% citrus; the Gunshop Fizz is 33% mixer and juice, 33% spirit, 17% syrup and sugar and 17% citrus. That makes them close cousins on the ratio map.
The Gin-Gin Mule is the stronger drink in the glass (10.4% ABV against 7.7%), but being smaller it holds about the same alcohol (1.18 standard drinks each). The Gunshop Fizz has more calories (about 240 against 205) and more sugar (29.5 g against 21.9 g). Both are shaken and served in a highball.