I think the main difference in rivalries between FIRST and conventional sports isn't the inclusion of GP or coopertition; it's simply a structural issue. We play so many teams in such a short amount of time for only 2-minute matches. The stands aren't split across the field between two rivals; they're filled with people from dozens of different teams who may or may not know each other. For the vast majority of our playing time, we don't know who we'll play, when, or with whom before the last day or two. There are exceptions, primarily based on geography (rival schools or colleges), heritage (same or friend/family mentors) or history (previous elimination play)--but these are inherently rarer.
On the other hand, it also creates a unique situation in which two good friends and/or hopeful elimination partners may meet across the field during quals. It's always fun giving a good trouncing to your friends or trying to shut down someone you're hoping will pick you. This is the most common "friendly rivalry" I see--at least a few per event--and it's definitely motivating!
You can also do stuff like
this.
148 is professional at all times...no one said you couldn't be a professional prank-puller.