Demandred has essentially the same motivation, but much, much stronger- he was born (birth name Barid Bel Medar) one day after Lews Therin Telamon (aforementioned prior incarnation) and his life followed that pattern from there on, with him being second to Lews Therin in almost every way, including skill with magic, political success, good looks, and even height. They even both courted the same woman, but she preferred Lews Therin. Then the War of Power broke out, and Barid Bel found out he was better than Lews Therin at one thing- war. Unfortunately, the powers that be made Lews Therin their top general anyway, and that final straw snapped Barid Bel's mind completely so that he became obsessessed with killing Lews Therin and joined the Shadow to better accomplish it, changing his name to Demandred in the process and quickly becoming its most feared general note Though the joke was on him still; the Dark One already had a Dragon, so Demandred was still stuck playing second fiddle, this time to Ishamael. In the last book of the series, Rand (who now has memories of his life as Lews Therin back) reflects on how he encouraged this rivalry and expresses regret that he drove Demandred to madness and evil when he could have been one of the greatest heroes of the age- had he not been born at the same time as Lews Therin.
A light-hearted comedy example can be found in the long-running "feud" between comedians Jack Benny and Fred Allen. The two would frequently appear on each others shows to harass one another and trade barbs, leading to a Crowning Moment of Funny when Jack Benny, who had taken an assumed name to appear on Allen's new game show, finds out his "big prize" is to have his pants pressed — right then and there.
Jack Benny: You haven't seen the end of me, Allen! Fred Allen: It won't be long now!
In Phoenix Wright: Ace Attorney, we have Miles Edgeworth, who was a childhood friend of Phoenix, but is now his rival. He eventually becomes Phoenix's friend again. Phoenix himself says that one of his primary motivations for becoming a lawyer was so he could face Edgeworth in court.
Franziska von Karma steps up to the bench for the second game. She violently wants to defeat Phoenix because she wants to succeed where Edgeworth failed.
She also shares this position with Shi-Long Lang in Ace Attorney Investigations, specifically in the flashback case four, where she's a thirteen-year-old obsessed with beating Edgeworth in everything. Edgeworth mostly ignores it.
And for the third game we have Godot who really hates Phoenix's guts and won't explain why. He blames Phoenix for Mia's death.
In Apollo Justice we have Klavier Gavin, the brother of Apollo's former mentor. He is way more helpful than the other rivals though.
Yanagi and Hiroki have this relationship in Canvas 2 after reconciling.
Battler has two through the course of Umineko no Naku Koro ni. On the first four arcs his rival is most likely Beatrice; while Battler tries to prove that all the murders happened by human methods, Beatrice opposes him stating that all were made with Magic, and they fight for a last time at the Hidden Tea Party of the fourth episode, with Battler victorious over a resigning Beatrice who already got tired of the game. On the Chiru arcs his rival turns out to be Furudo Erika, a piece placed by Bernkastel to serve as the detective, effectively hijacking Battler's role. Both clash numerous times and their conflict finally ends in the famous chained closed room deathmatch.
Captain Gamer's nemesis (Czar Gamer) regularly swings from friendly rival to bitter enemy and back again.
Bun-Bun also has a Rival in the form of Santa Claus, who tries to kill him each Christmas. An odd version of the trope, since Bun-Bun is a sociopath, and Santa is generally portrayed as a decent (if slightly obsessed) guy just trying to see that justice is done.
Lance has a rival in Gold Coin Comics named Blake, who has traits from Seifer in FFVIII.