My Hero Academia is recognized as this to Naruto. The author, Kohei Horikoshi, stated various times to be a huge fan of Shonen Jump, particularly Naruto.
Keiko Takemiya's Kaze to Ki no Uta is the spiritual successor to "The Door To Summer", a one-volume manga she wrote. They both contain similar story elements, as well as both being adapted into 60-minute OVAs.
Both of these series in turn are spiritual successors to In the Sunroom, a lesser known manga of hers, which also have similar story elements of the two series mentioned above.
Life is a succession to the mangaka's work Vitamin, as both deal with the harsh reality of bullying and how it can physically, emotionally and mentally destroy the victim. The differences are that the former is a long-running series spanning over twenty volumes, takes place in high school and the girl cuts herself and the latter is only a few chapters long, takes place in middle school and the girl becomes bulimic. How they deal with the bullying in the end differs, too.
Soul Eater is sometimes considered to be the spiritual successor of Fullmetal Alchemist. Note that Soul Eater's writer Atsushi Okubo has worked as Hiromu Arakawa's assistant.
The animated adaptations of Soul Eater and Fullmetal Alchemist even suffered similar fates. Both shows started to overtake the manga on which they were based on, and rather than try to pad out the show with filler, the writers came up with their own ending for the show to tie up the loose ends.
Brynhildr in the Darkness can be considered a successor to Elfen Lied, seeing as they are both written by Lynn Okamoto, contain cute supernaturally empowered girls who are mistreated in the name of science, gory fights and have harem undertones.
Tōka Gettan is a Spiritual Successor of sorts to Yami to Boushi to Hon no Tabibito and Moonlight Lady. One episode of Tōka Gettan was essentially a YamiBou episode. The games they were adapted from were unrelated, though.
Kannazuki no Miko has occasionally been cited as a Spiritual Successor of YamiBou, but that probably has more to do with their typical Girls' Love character designs.
The Big O could be seen as a successor to Daitarn 3, only using Batman instead of James Bond.
The Big O itself has a spiritual successor in Tiger & Bunny. Both are anime series by the same studio, share part of the staff, and has the look and feel of an american saturday morning cartoon. While the first sported a mecha pilot Bruce Wayne, the second depicts the results of a government/private take-over on the Justice League. Also including a cute short-haired android voiced by Akiko Yajima.
Ninku shares many similarities with Naruto despite coming first. Examples like the fact that Fuusuke is a wind ninja who wears orange and uses the kuatsuken (resembling Rasengan).
Fairy Tail is this to Mashima's earlier series Rave Master, containing a few characters with the same name and/or design, and even a few plot elements with the same name but different functions. Etherion, Oracion Seis, etc.
It's sometimes considered one to One Piece as well.
Durarara!! is this to Baccano!, although the two are different in tone and setting, they share the same creator, the same "Loads and Loads of Characters with distinct plotlines on a collision course" writing style, and the same jazzy musical style.
They also happen to share the universe, though taking place in different countries and decades apart, only two characters from the first make a brief cameo in the second.
Tamayura is the spiritual successor to Kamichu!. Not only is the show about a tightly-knit group of teenage girls, with the mood, art style and theme practically lifted from its illustrious predecessor, but it also takes place in Takehara, an old town not far from Kamichu!'s Onomichi and similar to Onomichi in many respects. Still, an argument could me made for Sketchbook as an influence, considering the age of the characters and the art theme (not to mention the opening).
Nodame Cantabile is seen as the spiritual successor to Honey and Clover. Both are about students in art schools (drama ensues, obviously), both anime adaptations were made by the same studio, with a very similar drawing style.
Your Lie in April is seen as a successor to Nodame Cantabile due to also being a series about music that has a prodigious yet cynical male pianist whose life changes when he meets a talented and wild female musician.
At the exact same time, Spiritual Antithesis is also in play. While sharing both racing themes, Wangan Midnight focuses more on street racing while Shakotan Boogie puts more emphasis on Running Gag and Shakotan-styled cars. Wangan Midnight also took the races on the expressways while Shakotan Boogie focus the races on touge and city streets.
And now, Wangan Midnight and its C1 Runner sequel were succeeded by Ginkai no Speed Star, which has a completely fresh new plot with automotive mechanics as its main focus, though it was set in the same universe and had the Wangan Midnight Final Chapters as its subtitle.
Spotted Flower is one to Genshiken, both by the same author and focusing on the otaku culture, although there are subtle hints pointing that the former might be a true Sequel to the later, with the two of the main characters grown-up and married.
Guilty Crown: General consensus is that this series was a partial one to Code Geass, particularly in the first few episodes, given that it shared the same "resistance group taking on oppressive entity" premise. While Shu and Lelouch have very different personalities and interests, they are both 17-year old youths accompanied by mysterious girls (Inori and C.C. respectively, who also differ from each other) and capable of using mysterious powers. However, the overarching story eventually headed in more of a different direction.
Dusk Maiden of Amnesia can be considered this to Natsu no Arashi! in animated form at least - both series were directed by Shin Oonuma and feature cute (actually, near-identical) ghost girls and a healthy dose of SHAFT-like visualsnote though only Arashi is a SHAFT production itself - Dusk Maiden is produced by Oonuma's spin-off studio, SILVER LINK.
In addition to having the same creative team, both Kill La Kill and Gurren Lagann share similar stylistic elements and cover similar themes. Where the latter is about boys growing up and learning to face the challenges of life (told though over the top Super Robot action), the former is about girls growing up and learning to grow past the expectations and pressures of society by way of Magical Girl Warriors wearing sentient school uniforms.
Kamisama Kiss has a lot in common with InuYasha. To begin with both plots revolve around an ordinary teenage girl getting control over a supernatural asshole, who is forced against his will to help her and protect her, and only later do the two start to fall in love. Tomoe, the male lead in KK, looks almost exactly like an older and more mature version of Inuyasha when he wears his hair long; he also acts like a more intelligent and slightly more mature version of Inuyasha. Nanami, the female lead in KK, looks like Kagome with brown hair and has the same personality type. Then the Time Travel Arc happens in KK and the similarities get even more pronounced.
Kuroko no Basuke, the most popular basketball series of the 2010s, is the Spiritual Successor to Slam Dunk, the successful basketball series of the 1990s.
Captain Earth towards Star Driver. It helps that both series feature the same staff of people (except for character design) and concepts from the latter show are used again in the former, most prominently the usage and explanation of the term "libido". Since Captain Earth is still running, the jury is still out wether both series are set in the same universe or not and speculations are running rampant. What can be said already, however, is that Captain Earth has an overall more serious tone than Star Driver did.
Sword Art Online to .hack. Both series center around Virtual Reality MMORPGs that are more dangerous than a video game ought to be, with SAO killing players who die in-game and .hack//'s "The World" housing dangerous AI experiments. Both also have video game adaptations that simulate MMO gameplay in a single-player Action RPG.
Yuri!!! on Ice to the Japan Animator Expo short ENDLESS NIGHT, both of which were directed by Sayo Yamamoto. ENDLESS NIGHT's primary focus (figure skating), core concepts, and some of its imagery (most notably that of two men pair skating together) are all transferred into Yuri!!! on Ice.
Space Patrol Luluco can be considered a Studio TRIGGER version of TYPE-MOON's Carnival Phantasm: both shows are comedies based on taking several different independent properties owned by the company (Inferno Cop, Kill la Kill for TRIGGER and Tsukihime, Fate for TYPE-MOON) and crossing them over with comedic results, along with being a sort of Self-Parody of the companies producing the anime.
One Piece is now one to Kochi Kame, not because of their content but by the fact that after Kochi Kame ended, One Piece has become the oldest series that is still serialized in Weekly Shounen Jump. As of 2017, it's celebrating its 20th anniversary and the only other series that premiered before the 2000 era is Hunter × Hunter, a manga that is barely present in the magazine due to its high amount of hiatus and it was even completely absent in 2015. The only other Long Runner in the magazine is Gintama which premiered in 2003 and every other series premiered in 2012 and later. One Piece has been serialized for half as long as KochiKame's serialization and it is very clear that One Piece isn't going to end soon.
While many shonen manga running in Jump and other magazines are heavily influenced by Dragon Ball, One Piece, more than any other series, is seen as it's generation's equivalent to that manga, having reached the same level of ubiquity and success in Japan, critically and financially. In an inverse of how it compares to Kochi Kame, it's run much, much longer than Dragon Ball but has similar content and characters.