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.
I admit there's some weirdness that they don't face off directly, but that's why.
Ah, politics. It's all perspective.
I considered typing out a novella, but it wouldn't mean much if you're as unfamiliar with the sport as you seem to be.
That said, American football is altogether inferior to the real football.
This is the critical distinction.
Peyton isn't deciding the defense when Brady is on offense. Brady isn't reacting to whatever Peyton left on the pool table or the dart board.
I think much of the same is true of pitchers in baseball as of quarterbacks in football, except pitchers only play something like 1 in 6 games, and these days it's very rare for them to pitch a complete game.
By your logic golfers could never be rivals, because they're never interacting or defending against one another -- they're just playing against the course.
This is the critical distinction.
Why is the commentator talking about Brady vs Peyton, when actually Peyton is doing nothing to stop Brady getting touchdowns?
Why don't they focus on whoever is making the decisions on the defensive side? That would seem to matter more as a struggle between combatants. It's not like each defense is some nameless, generic unit that simply reveals who is the better QB.
There might be some depth to the attack vs defense strategy that is worth highlighting, above the QB v QB.
For example, academic rivals compete in the classroom to have a better GPA, business rivals compete to win more customers and sales, etc.
Why isn't the rivalry considered to be between the QB and someone on the defense? There's actually two matchups in an NFL game (plus specials but whatever), the two offense versus defense pairings. It's odd to make the rivalry about two guys who aren't directly tackling each other, when there are people on both teams who really are tackling those guys.