Let's start by addressing the 3600 ft., 18,000,000 ton, transforming city/starship/robot in the room; What is an Anime/Manga themed Role-Playing Game?
Simply put, it's a tabletop RPG game - be it a single adventure, a system, or a campaign - that has features inspired by Japanese Manga (Sequential Art - Comic Books) and/or Anime (Animated Cartoons on TV or in Film format).
This can range from the types of characters, technology, monsters, and such that appear in these mediums to stories made in the style of Japanese pop culture fiction. These stories often feature genre conventions and tropes quite different from those typically seen in the majority of similar productions in the West.
So far so good, yes? Even if the above explanation is a tad academic, it's good to begin with a straightforward and concise understanding of what we're going to be talking about. The thing is, the information above is probably not anything you don't already know. The key question isn't 'What is an Anime/Manga game?' but 'How do you make a game feel Anime/Manga inspired or themed?'.
One of the first things to come to terms with is that Anime and Manga aren't really genres. They are a means of depicting a genre with stylistic elements specific to those mediums. Mobile Suit Gundam (the original anyway) is a Science Fiction War Story. Ranma 1/2 and Urusei Yatsura are Fantasy/Sci-Fi Romantic Action-Comedies. Record of the Lodoss War and Goblin Slayer are Western Fantasy. All of these are Anime/Manga however and share some elements tied to those particular means of telling stories.
Japanese cultural components appear in many Anime and Manga stories and they can be both confusing to the Western viewer but also really helpful in enforcing the feel and atmosphere of the medium. At the same time, I am not going to go into all those details here. This post is directed at the broad strokes. At the same time, I am going to give alternatives to many of the assumed cliches.
The Heroes Are Young
The protagonists of most Anime/Manga stories are often pre-teens to teenagers, even if their age would - in the minds of most Westerners - make their role in the story seem illogical. For example, Ash Catchem (Satoshi in Japan) of Pokemon fame is only 10 years old when he begins a cross-country journey all by himself to catch Pokemon and become a championship trainer. The pilots of the EVA units in Neon Genesis Evangelion are all 14.
The main reason for this is the same one that introduced teen sidekicks to Comic Book Superheroes in the West - the audience is generally comprised of kids who are in the age groups of the main characters and want to see themselves in the heroes and their universes. Simply put, a series with 12-15 year old protagonists is likely aimed at 12-15 years olds.
Of course this is not always the case and indeed wasn't the case at all in the 'Silver Age' of Anime and Manga. Captain Harlock, The Dirty Pair, Lupin the III, and many other classic characters are very much young adults to full on grown-ups. Likewise, the base audience for Evangelion tracked much older, as many of the themes in that series would not be familiar to high school age children.
When running an Anime/Manga game based on the current style of the medium, assume your PCs will be between 8-14 unless the setting calls for them to be older. It's not unlike Tales from the Loop and Things from the Flood in that regard.
The Setting Dictates the Specifics
Related to and expanding the aforementioned particulars of age, there is often but not always a setting reason why the characters are kids. Perhaps children has an imagination and psychic resonance we lose as we group up. Maybe they are the only one's presently conscious when a giant alien monster attacks and the experimental mecha tuned themselves to the teens when they give into them; unfortunately for everyone involved they won't work for anyone else now. Why are the kids in My Hero Academia so young? Why are the Harry Potter characters so young? Dude, it's a story about people in middle and high school!
Also of note regarding the mandates of setting is the autonomy of the PCs and whether or not they are treated like normal young people of their age. The characters of My Hero Academia are in high school, have to attend classes and school events, and can't use their powers to fight crime yet unless it is approved by senior, pro-heroes. On the other hand, Ash, Misty, and Brock (Satoshi, Kasumi, and Takeshi) can wander the Sinnoh Region, a land roughly the size of the Japanese island of Hokkaido (which it is patterned after) completely unsupervised, going where ever they want and doing whatever they wish.*
GMs should consider what their campaign world deems 'normal' and convey that to the players. Again, and I can't stress this enough, not what Our Real World deems normal but what is standard practice for the setting.
It's Not About The Giant Robot
Unless of course it is...but more on that later.
The main thing to remember about Japanese pop culture entertainment is that mixed into the Giant Robot battles, Magical Girls fighting an ancient darkness, and Demon Slaying Samurai is that it is all the backdrop for the real story; Anime and Manga tell tales of friendship, romance, determination, duty, the futility of war, environmental themes, and other emotional and philosophical concepts closed connected to not only Japanese culture but the Human condition.
There is a reason Japanese Anime and Manga, a thing wholly and strongly tied to a particular place and people, has become beloved by fans the world over. Almost regardless of who you are and where you're from you are bound to understand the ideals being fought for by the heroes of this medium. We all love, we all get jealous that he like her more than us, we all want to fight injustice and protect those we care about, and we all love cute creatures that can zap our enemies with lightning bolts. OK, for some that last one is stronger than it is for others.
The point is, your Anime/Manga campaign has Giant Robots or it has four and a half foot tall, teenage elf girls with six foot long swords. It isn't about those things. It's about people. It's about the characters, their belief, desires, goals and their situations. The rest of the tropes are there to make it 'look' cool in your head.
More to come...