The war that started in September of 1939 was called World War II even before it had truly become a world war. It was first called that in Time Magazine of September 11, 1939. At that point, the war only included Germany, which had invaded Poland, and Britain and France, who had declared war when Germany did that but had yet to take any real action.
The reason that Time called it WWII was because the previous great war had come to be known as WWI and it seemed likely that this war would end up involving many more countries than were involved right at the beginning. There were so many points of potential conflict around the world that it seemed certain the war would become much bigger than it was.
And the world war involved couldn't be called a European war because Asia,Africa and some other parts of the world joined.Even USA joined indirectly because Americans liked isolation and didn't want to take part in the war but for its own interest it helped Britain in a way and Britain didn't have to pay back.
Also, WWII was an extension of WWI because the issues that started the first Great War were never really resolved. Plus, Hitler compounded on those issues by adding to them his own agenda (i.e. the persecution of the jews).