The whole "Team" thing originated years назад with the infamous "Team Aniston vs. Team Jolie" incident way back when Brad was still married to Jennifer.
It's just a natural thing. Choosing your team doesn't really have anything to do with Twilight. It's just if Ты like Gale more, Ты support Gale. And if Ты like Peeta more, Ты support Peeta. So Ты just say "Team _____" to let everyone know who you're rooting for. :) Saying Team ____ didn't originate with Twilight.