Actually the Golf was named after The Gulf Stream undersea current. The German spelling for 'Gulf' is 'Golf'.
This was in line with VW's naming convention of naming their cars after winds and currents. e.g. Scirroco, Bora, Vento, etc.
Somewhere along the line, the name Golf was associated with the game of Golf (pre-GTI).

The americans dubbed it the Rabbit as to differentiate it from the European spec Golf - a marketing thing.