It basically comes down to the face that the quarterback position has an outsized importance in the game and those players become the face of their franchises. They are often credited with wins and losses despite those being a team stat.

Some of the players do find this fan/media obsession strange. They may be asked about playing against QB X and reply that they're actually playing against the opposing defense.

There are rivalries between players who are on the field at the same time, but they're less prominent. Those between a wide receiver and cornerback are probably the most common. Throughout the game they are racing, pushing and deceiving each other and fighting over the ball and it can become heated.

However, it's also possible to have a competitive rivalry without directly facing each other on the field. In individual sports like golf this is the only way. But in other sports players can also compete with each other on some individual aspect. In baseball, the 1998 HR chase and Dimaggio brothers are some prominent examples. In Brady's case he had a strong desire to be number one. The threat of Peyton winning MVPs over him, having better statistics than him or Peyton's team winning over his was a strong motivator for Brady to protect his status.