In InuYasha, Byakuya comes in after Kagura's death and not only looks and acts like her, but has similar powers and plays the same role that she played.
In Naruto, Sai, the quiet, serious and emotionally distant ninja replaced Sasuke, the quiet, serious and emotionally distant ninja. It doesn't help that they look so similar, either. Much like Sasuke during the early arcs, Sai's rocky friendship with Naruto and Sakura has slowly eroded his stoicism.
The similarities between Sai and Sasuke have been lampshaded in-series. During his introduction this was played for drama, with Naruto and Sakura openly hostile at Sai attempting to replace Sasuke. Afterward it was played for comedy, such as Konohamaru's use of both Sai and Sasuke in his male version of the Sexy Jutsu.
Vincent Volaju is one for Vicious in the Cowboy Bebop movie.
The female coordinator character May (who replaced the significantly different Misty) was replaced by the female coordinator character Dawn.
Dawn was then replaced by Iris, who in a roundabout way, is actually closer to Misty in terms of personality.
Brock was replaced with Cilan, who is the same with more food focus and less lechery.
For the X and Y saga, we are introduced to Clemont and Bonnie, who are a lot like May and Max but with their birth order reversed, also Serena, who appears to be a bit like May or Dawn.
Jessie and James release their Victreebel (a carnivorous plant Pokémon) and Arbok (a poisonous snake Pokémon). Later on, James acquires a Cacnea (a cactus Pokémon) and Carnivine (a carnivorous plant Pokémon), which both retain Victreebel's tendency to accidentally injure its owner in displays of affection, and Jessie gets a Seviper, which is...a poisonous snake Pokémon.
Ash gets a new bird Pokémon in every region which eventually is fully evolved (Pidgeot, Noctowl, Swellow, Staraptor, and Unfezant respectively).
The Mewtwo that appears in Genesect and the Legend Awakened is a different one from the one in Mewtwo Strikes Back and its direct sequel. However, despite a slight change in personality, voice and abilities, it functions almost exactly like the original, and little justifies the existence of this new one (in and of itself a bizarre choice as prior to this it was universally depicted as a Single Specimen Species). Made even more apparent considering the prologue dedicated to it gives it almost the exact same past as the old one, but with all things Team Rocket related removed, and the film's ending, which puts it in a city... which is exactly where we last saw the original Mewtwo.
The character Priss from Bubblegum Crisis was originally going to be killed off and replaced by Vision, who was suspiciously similar (brown hair, kind of feisty, was a singer) but ultimately fans liked the character, so she wasn't killed off after all, thus avoiding the trope entirely.
Tenchi Muyo! is unique in that the third OVA did something of an alternate continuity substitution. Mihoshi's partner in the Galaxy Police had long been Kiyone Makibi in the Universe and Tokyo continuities, but supplemental materials Kiyone was the name of Tenchi's Mother (The first movie, based off the first TV series, named her Achika). OVA 3 introduces us to Noike, who happens to be Mihoshi's previously not known to exist GXP partner, while Tenchi's mother is finally officially named as Kiyone (and (re)introduced, in a way, by Tenchi's older sister who is strangely identical to her).
Near from Death Note. This is intentional — Near tries to imitate L and eventually admits that he failed to imitate L, but succeeded through Teeth-Clenched Teamwork.
The reasoning in the story being rather interesting, Near has more tendency to 'cheat' in his methods, the example being that if he were in L's shoes, he'd have released all information on the Death Note to the public and let the ensuing mass manhunt capture Kira. He tries to imitate L because he knows that in the real world, his methods would cause far too much collateral damage even if they succeeded.
Shikabane Hime's Keisei dies, allowing Ouri to come closer to Makina and to also be brought into the story as a major player instead of a male damsel in distress and hapless bystander. About 2 episodes later Keisei's mentor is introduced. He has a very similar hairstyle and replaces the former as the even more perverted comic relief and Ouri's new mentor.
In Mobile Suit Gundam 00Neil Dilandy, killed late in the 1st series, is replaced by his twin brother Lyle Dilandy.
After Tup Dop, a supporting character of D.Gray-Man was killed, he was replaced by his equally chubby and androgynous sister, Cash Dop.
In Patlabor, Takeo Kumagami replaces Kanuka Clancy after Clancy returns to America. They have very similar skills and fulfill the same police positions, although Kumagami is given a few quirks to distinguish her.
Each season'slancer is a Suspiciously Similar Substitute of Yamato from the first season. He, Ken/Digimon Kaiser, Ruki, Kouji, Touma, and Kiriha are varying degrees of blue oni to the Goggle Boy's red oni, each has some connection to the color blue, all but Ken and Kiriha are partnered with or transform into canids, and three of them are blond males. Kiriha's an odd duck, though, as he is partnered with a Greymon, traditionally associated with the Goggle Boy. At least it's blue this time around.