It can. But it's not why I chose it. I adopted that monicker when I was elected as the "social" chair ("anti-social chair" wasn't an electable office) of my grad school program at Berkeley. (Not that I ran for the office either -- as I was chosen for it while absent from a meeting when I was lost on a hillside near Danville, CA during an evening grad school offsite. But that's another story.)
Too many years ago, I came across a white sale catalog in the mail one day while trying to avoid studying (a JC Penney ad, in fact). In it, there were all sorts of "valances" for sale ... basically window dressings/curtains for your home. They each have different names for different styles. One was called a "Swag Valance", which was just too brilliant -- so I adopted it as my cheesy Holiday Inn lounge singer alter ego. Now a name like that said "Social Chair" to me. I've used it ever since.
Swag is just the short form.