Adidas does not stand for "All Day I Dream About Sports." That line is a backronym, invented after the fact, and Wikipedia lists it among the myths about the name. The real source is simpler. Adidas is a man's name, shortened.
The man was Adolf Dassler, known as Adi. Take his nickname, add the first three letters of his surname, and you get Adi-Das. He registered the company in Herzogenaurach, Germany, on 18 August 1949.
The name exists because of a family feud. In 1924 Adi and his older brother Rudolf set up a shoe business together. They called it Gebrüder Dassler Schuhfabrik, the Dassler Brothers Shoe Factory. For more than two decades they ran it side by side in the same small Bavarian town.
Their relationship broke down during and after the Second World War. Accounts of the cause vary. Wikipedia's article on Puma describes a misunderstanding during a 1943 Allied bombing raid. It also says Rudolf came to believe Adi had betrayed him to American soldiers. Sources give the final split as 1947 or 1948. Either way, the brothers divided the business and never reconciled.
Each brother then did the same thing: he named a company after himself. Rudolf chose Ruda, from Rudolf Dassler. He soon changed it to Puma, registered in 1948. Adi made Adidas from his own name a year later. The two rival brands began as mirror-image abbreviations.
The feud split the town too. Rudolf set up across the Aurach River from his brother. Herzogenaurach became known as "the town of bent necks." People looked down to check which brand of shoes a stranger was wearing before deciding how to treat them.
The three stripes came later, and they were bought, not invented. In 1952 Adidas acquired the stripe trademark from Karhu, a Finnish sports company. Wikipedia reports the price as about €1,600 and two bottles of whiskey. Britannica notes that the brand has been styled in lowercase, "adidas," since 1949.
So one of the world's best-known sports brands is, at root, a signature. It carries the founder's childhood nickname into every stadium.