Microdisney were like this, with Cathal Coughlan being the red oni and Sean O Hagan being the blue oni. Their music was known for the contrast between Cathal's angry/melancholic lyrics and Sean's euphonic, peaceful melodies. This became even clearer when the band broke up due to a clash between the onis - Cathal wanted to make more aggressive music, Sean wanted to make softer music.
Queen: Freddie Mercury and Roger Taylor were Red Oni, while Brian May and John Deacon were Blue Oni.
Space had Jamie Murphy and Tommy Scott as the respective oni. Jamie was a Fiery Redhead whose constant partying caused him to have a nervous breakdown, while Tommy was the more sensitive and introverted one of the two. The trope also applies in the musical sense as Tommy's songs were more melodious and romantic - albeit in a rather twisted way - while Jamie's were darker in tone, particularly on Spiders, and more indie rock-oriented.
During the heydey of Britpop, Oasis (the loud, outspoken, 70's rock and Beatles-influenced working class band) were the Red Oni and blur (the quieter, witty, Kinks-influenced band consisting of University intellectuals) were the Blue Oni. Because of their respective Oni characteristics, Oasis were the more popular of the two, while Blur had more critical acclaim. The two bands also hated each other for most of the 90's, and so did their hardcore fans.
In The Who, Roger Daltrey was a more reserved, polite musician while Keith Moon was the Crazy Awesome drummer. They didn't always get along off-stage.
Aerosmith: Steven Tyler is the loudmouth, flamboyant frontman (Red) while Joe Perry is the quiet, mellow guitar player (Blue).
ABBA: playful and exuberant Anni-frid Lyngstad was the red oni, while discreet and reserved Agnetha Faltskog was the blue oni.
Bon Jovi: All-round nice guy Jon Bon Jovi is the red oni; and the more difficult, for lack of a better word, Richie Sambora is the blue oni.
Sleater Kinney were an aural equivalent of this trope. Corin Tucker, whose voice was often harsh and abrasive, was the Red Oni. Carrie Brownstein, whose voice was generally much smoother and gentler, was the Blue Oni.
The post-1999 Red Hot Chili Peppers had the loud and over the top Anthony Kiedis and the quiet, thoughtful John Frusciante, who got noticeably calmer and more reserved when he re-joined the band. Of course, his successor Josh Klinghoffer can fit the Blue Oni role as well.
Kix Brooks and Ronnie Dunn, respectively, of Brooks & Dunn. Kix was the more active and energetic of the two in a live setting, while Ronnie was more straightforward and less showy.
Likewise, Jennifer Nettles and Kristian Bush of Sugarland. Nettles is very active and outgoing onstage, while Bush is more quiet and in the background.
The music video for Wiley ft Chew Fu's Take That features two dancers, Shanie Ryan and Yasmin Ogbu. Throughout the video, Yasmin (Red) pulls many crazy facial expressions and mouths some of the song, while Shanie (Blue) retains the same neutral expression (save for the occasional smile) and doesn't mouth any of the words. Ironically, in closeup shots, Yasmin is cast under a blue light while Shanie is cast under a red light.
The Beatles has two Red Oni, Blue Oni pairs. One is charming crowd-pleasing Paul McCartney and deep meaning-searching John Lennon. There is also cheerful Ringo Starr and quiet George Harrison.
King Crimson, from their 1980s incarnation onward, has a prototypical red-blue pair in their two guitarists, vocalist Adrian Belew and de facto leader Robert Fripp. Belew stakes his claim on raving, David Byrne-like monologues, bizarre guitar noises, and dancing around on the stage, while Fripp sits in the corner unobtrusively and plays immensely complex lines, while presenting a generally intellectual image.
Led Zeppelin: red Robert Plant's manic vocals and stage presence, blue Jimmy Page's steady and tight guitar virtuosity.
The Doors' Jim Morrison was definitely the red to all the others' blue. In a biography, Morrison is recounted as saying that keyboardist and band elder (by about 5 years) Ray Manzarek could give him a certain Disapproving Look and he'd know he'd "gone too far". But boy did he go pretty far a lot of the time....
Most lead singers are probably going to be comparatively red by virtue of them being the main stage presence and the voice of the band, and expectations of being particularly energetic in concerts.
The two major singers and songwriters for the Eagles, Don Henley and Glenn Frey, presented a red-blue contrast. Henley's rougher and raspier voice was generally put to use in their more energetic or "harder" ("Life in the Fast Lane", "Those Shoes"), or more emotional ("Desperado", "Wasted Time") songs. Frey's easier, smoother voice generally goes to the easy countryish songs ("Take it Easy", "Tequila Sunrise", "New Kid in Town"). Henley is red, Frey is blue. (And in their respective solo careers the contrast becomes more apparent.)
Comparing the big two "British Invasion" bands of the early 60s, the Rolling Stones and The Beatles, the Rolling Stones played red to the Beatles' blue in their little rivalry. Even as the Beatles got harder and edgier in the late 60s, the Stones continued to be on the red side of them.
Heavy Metal has many subgenres that form this dynamic with each other; a well-known example being the dynamic between Black Metal (Blue; focused on atmosphere and ambiance) and Death Metal (Red; focused on aggression and intensity).