Michael Schumacher (red) and Mika Häkkinen (blue) in Formula One.
Vince Lombardi and Tom Landry (Red and blue respectively) coached one of the greatest games in American Football history against each other, the 1967 Championship game. Lombardi was described as a very impassioned, hot-blooded coach who would celebrate every success and and mourn every setback with equal gusto, whereas Landry was considered a very calm and stoic coach. Ultimately, Lombardi's Packers beat Landry's Cowboys 21-17. They also played the same red/blue roles when together as assistants for the NY Giants - Lombardi coaching the offense and Landry coaching the defense.
Shaquille O'Neal (red) and Tim Duncan (blue)
Karl Malone was the blue (blue-collar work-ethic, more reserved demeanor) to Charles Barkley's red oni (party animal, loud, very outspoken).
Martina Navratilova and Chris Evert had a long-running Friendly Rivalry in the 1980s with Navratilova being the more openly emotional and passionate one and Evert gaining the nickname of "Ice Queen" for her cool and steely demeanor on court.
The choleric John McEnroe and the imperturbable Bjorn Borg had one of the most famous tennis rivalries in spite of its relatively short length, mainly because of their perfectly contrasting "fire and ice" personalities. Additionally, the robotic-seeming Ivan Lendl could be said to be a replacement blue oni for McEnroe after Borg's retirement.
Venus Williams (blue oni) and Serena Williams (red oni)
The giggly, bubbly Monica Seles was the best-known and most successful rival to the cool-headed, stern-lipped Steffi Graf.
USC and UCLA. The Trojans in cardinal and gold tend to be more proud and arrogant as the red onis, while the Bruins in True Blue and gold tend to be more intellectual and nerdy.
City rivals Rice (blue) and Houston (red) fit this trope to a T, in both their student body profiles AND their respective choices of school colors.
The University of Georgia and Georgia Tech. The colors even match, as Bulldogs (in red and black) are rough and rowdy sports fans while the Yellow Jackets (in white, gold, and navy) had a reputation for being intellectual "engine-nerds."
Boston Red Sox (blue collar, ironically) and New York Yankees (metropolitan)
Like Dr. Manhattan, everyone is a hot-blooded underdog compared to the New York Yankees.
Anaheim Angels and Los Angeles Dodgers (just for color, they both used to be blue)
The Washington Capitals and Pittsburgh Penguins, embodied by the playing styles and personalities of their respective superstars: the Caps' Alex Ovechkin (red) and the Pens' Sidney Crosby (blue). Appropriately, Washington's primary color IS red, while Pittsburgh used blue as its primary color until 1980 and recently brought it back for their alternate jersey.
Eternally present in European Football Clubs, especially as most local rivals tend to wear Red and Blue to contrast each other intentionally.
In Italy's Serie A, AC Milan wears red, and they're a passionate club, with a fanbase mostly from the lower class. Inter wears blue and their policy is calculating, with their devoted fans coming from the upper class.
Manchester United and Manchester City. United is the much more fiery one, given their huge fanbase (accounting to 5% of the world's population) and the number of silverware they accumulated over the years. City is much more closer to the upper class Mancunians and residents of the city proper, in contrast to United's wider demographic.
Arsenal and Tottenham Hotspur, also known as the North London Derby. Arsenal is Red, with a very diverse multiethnic fanbase. Tottenham is Blue and is supported by the conservative Jewish population of London.
A teammate example: Italy's and AC Milan's midfield duo, the fiery, aggressive Gennaro Gattuso and the cool, efficient Andrea Pirlo.
Another one: the Paris St Germain strike pairing of the arrogant, explosive Zlatan Ibrahimovic and the calm, methodical Edinson Cavani.
Another teammate example: Chicago Blackhawks stars Jonathan Toews and Patrick Kane. Together the two are the face of the team and, to an extent, the NHL as a whole. Kane tends to be a bit more excitable both on and off the ice while Toews exhibits a much more cool and collected personality.
In British football, Luis Suarez, the fiery Uruguayan with a penchant for spectacular goals and a knack for being controversial, is the red oni to the blue oni of Sergio Agüero, the methodical Argentine who's a cold-blooded, clutch finisher with sound fundamentals. This becomes even better when you look at the jerseys of their clubs. (Luis Suarez - Liverpool, Sergio Agüero - Manchester City)
The three major Texas colleges fit a slightly mutated version of this trope. The University of Texas is the high profile, upper class, some would say snobbish blue oni. Texas Tech is the hot blooded, unsophisticated, very red scrappy. And Texas A&M is kind of the purple middle man, who can fill either role depending on which of the other two they're facing next..
NFL team the Pittsburgh Steelers have a pair of outside linebackers who sort of fit. LaMarr Woodley is the Red Oni and James Harrison is the Blue Oni. LaMarr is more talkative, more likely to celebrate (every sack he records is followed by the most awkward flying side kick ever witnessed), and generally more energetic. James is a quiet, not-quite-right-in-the-head fellow with a penchant for perpetrating unspeakable violence with little noticeable emotion.
The two American Football teams in the NYC area: The Giants tend to be the Blue Oni (reserved, hard-working players in a clear-cut meritocracy where production on the field is all that matters) to the Jets' Red Oni (even though the Jets actually wear green, and the Giants have more red in their color scheme than the Jets do), who tend to have a lot of bluster.
This came to a head in 2011 when both teams faced each other on Christmas Eve: the Jets talked a lot of trash before the game, even covering up the Giants' mural of previous Super Bowl championships (both teams share the stadium, but as the designated "home" team, the Jets had control over the stadium itself). The Giants didn't respond much to this... at least in the media. On the other hand, they completely destroyed the Jets, ending their playoff hopes and starting the Giants on a journey to their fourth Super Bowl title.
All of the things said about Carroll could be said about Jim's brother John Harbaugh. John and Jim coached opposite of eachother in Super Bowl XLVII and the difference in temperament was astounding.
The Seattle Seahawks are known for 2 players. On one hand you have the brash, outspoken, hype-machine that is cornerback Richard Sherman. On the other hand you have the calm, cool and collected quarterback known as Russel Wilson.
The Calgary Flames and Edmonton Oilers of the late '80s were an inversion. The blue-coloured Oilers were young and high-scoring, led by the likes of Wayne Gretzky, Mark Messier, and Jarri Kurri, while the red-coloured Flames played more of a defensive, tight-checking game.
The University of Arizona and Arizona State University are both red-ish in both school colors (Arizona's is cardinal red and ASU's is maroon) and reputation (both are party schools) but Arizona is considered to be a better academic institution and also has navy blue as its other school color, while ASU's second is gold.
Sports/sports culture in general: American football is red to baseball's blue.
Hunting is red to fishing's blue.
In Indy Car, the very jovial, talkative, and friendly Helio Castroneves is the red oni to the much more calm, stoic, and businesslike blue oni of Scott Dixon, who beat him out for the 2013 championship. Ironically, Castroneves's normal livery is a black and white one (with the occasional red and yellow or blue and white livery depending on the event), whereas Dixon always drives a red car.