In Slaughterhouse-Five, Billy is, quite literally, a man out of his time. Having experienced some sort of a time-slip, he believes he has seen his whole life, past and present, including traveling to a distant planet. While he is sheltering in a slaughterhouse during an air raid in World War 2, however, he is offered and accepts a fur-lined jacket in contrast to the other men's more austere military attire.

This is a ludicrous look and is very strange for anyone to wear in that moment, especially a captive soldier. The effect of him wearing a fur-lined jacket illustrates how bizarre of a character he is, and the displacement he feels from his current situation due to his experience envisioning the rest of his life. He feels out of place in time, and therefore is somewhat unfazed and unaffected by the events he's experiencing in real time, not registering how bizarre a fashion choice this jacket is and how absurd it makes him look.

The fur collar is part of what makes Billy's outfit so ridiculous. Everyone else is given a military coat, heavy duty and without frills. Not only is Billy's jacket less useful for warmth, it is also the clothing of a primadonna or fashion-oriented person. 

Billy is not concerned with fashion. Far from being interested in fashion, Billy is fully disinterested and disengaged with the events taking place around him at this time. 

The fur collar exaggerates the degree to which Billy is made to appear ridiculous and also emphasizes his disengagement. Other people, he is told, would have refused the jacket on the grounds that it is an insult.