Rudolf Dassler founded Puma in 1948 after leaving the business he shared with Adolf Dassler.
The Dassler brothers had operated Gebrüder Dassler Schuhfabrik, a shoe company in Herzogenaurach, Germany. Their partnership ended after a serious personal and business split. Adolf went on to register Adidas in 1949, while Rudolf established the company that became Puma.
Puma was initially known as Puma Schuhfabrik Rudolf Dassler. The rival companies remained based in Herzogenaurach, where their competition influenced local workplaces, sports sponsorship, and even consumer loyalties.
Adidas is therefore the key common mix-up: it was founded by Rudolf's brother Adolf, not Rudolf. Nike and Reebok have entirely different American origins and were not products of the Dassler family split.