No, it isn't for looks... or club... I have one on all my bikes all the time. Serves several purposes. 1st, to clean my glasses. It also comes in handy to stop the flow of water when it's raining that comes off the front wheels (no front fenders). It's occasionally used to cover my face when it's cold... sometimes it's tied to a lock and jammed in my rear pocket (depending on where I'm at and who I'm around and who I'm with...) My club colors are maroon and white but my bandanas are always blue. The reason should be obvious...
and yes, they get faded and raggedy pretty quick so they get replaced often...
I do it for the times I stop at a gas station that doesn't have paper towels available, and the gas spills a little on the tank. Easy to get to, and always there.
I normally carry my bandana in my rear pocket. If it is on the handlebar it means I used it to wipe up some water and did not want to put the wet bandana back in my pocket so I tied it to the handlebar to dry.