Whew, sorry, this turned into a bit of a text wall! But I figured writing about this could be a good means of shedding some light on the extended canon of Sundown society. Hope you enjoy!
Sundown is kind of an allegory for mild British Wartime socialism (HAHA I TRICKED YOU ALL THIS FUN ROMP THROUGH THE ROOFTOPS IS ACTUALLY SYMBOLIC) so encouraging competition between the teams doesn't really make sense. There's already enough hostility between sectors (particularly in regards to Sector V, whom most people loathe both because it's lead by the son of the enemy and crewed by some of Sundown's youngest and oldest runners) without introducing a formal practice for it: the point of the sector system is to encourage community cohesion ('Sector V are good guys, they lay down their lives to protect us. Sector Y are good guys, they scavenge for the food we eat. Sector H are good guys, they heal our satellite settlements when the doctors in Sundown are too busy'), and because they all do such drastically different things (Sector E are a division of demolition experts who clear debris from lower chunks of the Sundown skyscraper for settlement purposes, Sector S are scouts who are regarded as the fastest runners around, but simultaneously are usually totally unfit for combat engagements, etc) there'd be no way of organising a competition to fairly gauge teams on equal footing. Put a medic up against any member of Sector X and they're gonna fall, that's less to do with their talents and more to do with where they're directed. That's why the brawl between X and Y was so controversial: because it kind of contradicts what the original elders had planned for Sundown at large, a community where the individual works for the good of the whole, as opposed to for fame and glory.
This, similarly, is why a runner is expected to lay down and die when captured. Outgoing squads (V, X, Y, H, N, S and the defunct O and Zed) are comprised mostly of people who are too stubborn to answer to the enemy and would sooner die out of spite than give in to torture: it's for the good of Sundown as a whole, the enemy can never know where to find them. Haha, you didn't think hard-heads like Church, Eva, Klaus, Marina and Melanie ended up on the same team by coincidence, surely?
This said (and I understand that you weren't made aware of this, that was my folly: the post that established it was before this iteration of RPG), the sectors kind of do have fan clubs. In a prior post, as Church was making his way to the dwelling of The Elders to meet his punishment for his team mate having killed in cold blood, he passed a wall on which I noted there were drawings of, amongst other things, Sector V. V, X and Y in particular are actually sort of civilian favourites, given they all perform such crucial roles: V is the first line of Sundown's defence, and their sword when it comes to stabbing out into the dark unknown that is New London. Likewise, Y is the reason they can eat, and X is the only Sector trusted to use martial force to keep the hordes from Sundown's door. They're all very popular, not in a "GO SECTOR V, GO SECTOR V" kind of way, but definitely in a "I want to grow up to be like them one day" sort of way. Plus, people do root and spin stories and legends out of Sectors facing their respective Omega Band (Sector V. vs. Band V., Roark's designated group, for example.)