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.