Players come and go, the fans and history is what makes the rivalry. Paul Pierce has said on many occasions that growing up in L.A. he hated the Celtics. I don't think any player at UConn generally hates Syracuse.
And if you're mentioning the 80's and 90's, Magic and Bird were great friends but were rivals on the court. Same thing with Ewing and Jordan. Best of friends in the off season until the season started. Melo and LeBron are the best of friends and have been playing each other since high school. Doesn't mean that Melo isn't aware of the fact that LeBron stands in the way of him getting out of the East for the next few years.
The players playing for the teams know that team's history with certain teams and know the difference when a rival comes to town over any other team. I remember prior to Knicks/Heat series the players were acknowledging the whole Pat Riley leaving NY for Miami and the wrestling/boxing matches that occurred during the games in the past.
Magic and Bird hated each other for years. The friendship came later. In their case, the teams the played on enhanced the rivalry, but it was more about the two of them. Their NCAA catch-up is still the highest rated Final Four game of all-time. Their rivalry was about timing, comparable skill sets, race, and talent. Hard to repeat any of that now. LBJ vs. Durant could be a great one. Only time will tell. LBJ is the better player, Durant has the better team.