One of Santonio Holmes' enduring legacies with the New York Jets is that, under Rex Ryan, the team no longer chooses permanent team captains. Although most teams elect captains, Ryan used to select them himself. After Holmes' fiasco against the Miami Dolphins, which involved getting ejected from the team huddle by his teammates, all while wearing a yellow C on his jersey, the team no longer engages in such a practice.
However, the team doesn't just ignore the position. They have two functions which essentially replicate the same purpose. First, the team has a "steering committee" of veterans, who are essentially the captains and guide the team from a leadership perspective. Second, Ryan chooses game day captains, which change every week. The purpose of the game day captains is symbolic and simply involves the coin toss, but it lets players feel special by allowing those who have a special connection to that week's opponent get highlighted.
@DWAZ73 I'd guess: Geno, Mangold, Wilkerson, Davis. And Bellore, if they did special teams captain.— Seth Walder (@SethWalderNYDN) September 1, 2014
So essentially, the team already has captains, even if they don't call them that. It doesn't particularly matter to me, since the team is going to have leaders anyway, but it's interesting to note the difference between the Jets and other teams in the league. As a side note, it's entertaining to me that E.J. Manuel was not chosen as a captain for the Buffalo Bills, but again, it hardly matters much. Who do you think would be team captains, if the team elected them?