With regard to soccer, the erstwhile Metrostars is now known as Red Bull New York in the vein of parent club Red Bull Salzburg, and the players are individually referred to as Red Bulls. I think in general with soccer, the team nickname is an afterthought compared to the formal name, like Manchester United and Chelsea FC. I'm sure they have some sort of nickname, like the Barons or the Tigers, but it's not that big.
And here I was thinking they changed the name to the Red Bulls because the company that owns Red Bull (yes, the "IT GIVES YOU WIIIIIIIINGS!" Red Bull) bought the team..I mean their logo is the freaking Red Bull logo off the can..
Oh, I see they bought that other team too. Maybe they should buy the Chicago Bulls while they're at it.