I guess it needs to be said that no, these are not the only types of characters you will ever see and no, it isn't impossible to twist a cliche and turn it on its head. In fact, I am not detailing anywhere near the complete list of every standard character type seen throughout the history of Anime. That would be ridiculous, making this post ten times longer than it already will be. These are simply the types of characters you see most often that lend themselves to being used as Player Character in tabletop RPGs.
I would argue that the best Anime/Manga characters are actually some combination of these archetypes. Furthermore, while I hope this post is helpful, the best way to get to know the types of characters you encounter in Anime and Manga is to watch Anime and read Manga.
Your mileage my vary.
The Anxious Hero Character: This is a very popular character type in Anime and Manga that, while not uniquely Japanese, is certainly seen more often there than in traditional Western Adventure stories. In the US and elsewhere, the hero is universally brave, tough, and confident. He (it's usually a he isn't it?) gets by on his good looks, mean right hook, and being good at practically everything he does exactly when he needs to be.
The Anxious Hero is the opposite of this in many ways. This character lacks confidence, isn't particularly good at anything at first, might be clumsy as a result of being nervous, doesn't want to fight, and often has not desire to get involved in the craziness going on around them at all. By interacting with the other characters they develop friendships, a much needed sense of confidence, and the ability to pull through and save the day.
I would say from my own personal experience as a GM, this is the toughest one to see realized at a gaming table. Western gamers, generally speaking, have trouble playing a lack of confidence. They never want their characters to appear frightened, unsure, or unprepared. If someone gets it, gets the concept of this character, then they will usually shine in an Anime/Manga themed game.
There are numerous variations of this character and for those in the Anime/Manga know, both Shinji of Evangelion and Deku of My Hero Academia show traits of this character type, yet those are two very different characters.
The Arrogant Character: Referred to in Japan as the Himedere (female) and Oujidere (male), these characters see themselves as Princesses and Princes and expect everyone to treat them as such. Common traits attached to these types are them being actual royalty or nobility, exceptionally wealthy, the children of well connected and/or famous people or being famous themselves (they are actors or pop stars). A variant on this is the Kamidere or 'The God Complex Character' who believe themselves exceptional in all ways and superior to everyone else just because they are them. Both of these character types are more commonly villains than heroes but in some instances they may be seen as anti-heroes (such as the main character in the series Death Note).
The Cynical Character: This character is cold, hard-hearted, blunt, and sometimes may include traits such as arrogance and a biting sense of sarcastic humor. Other times, the character is more quiet and aloof. Their cold demeanor can make them seem emotionless and unfeeling but in truth they are shielding themselves and hiding a deep caring for family, friends, and their significant other or crush.
A notable variation on this archetype is the Hinedere, whose cynical world view comes with a high level of arrogance.
The Disciplined Character: Serious, focused, no nonsense and often staring down their nose at the more undisciplined in the group, this character tends to have dark hair, glaring dark eyes, and glasses are not uncommon. Proper discipline in all things is key with this character. Here is someone who knows the proper procedures, protocols, and rules and follows them. In school settings this is your Class President, often a member of the archery, fencing, or kendo club. Chess is usually their game of choice most.
A variation of this is The Dedicated Character who shares many of the attributes of the Disciplined one but is more focused on a single thing. A Scientist, a Corporate or Government Middle Management person, and the Captain of the Swim Team are all good examples of this character archetype.
Sometimes, though not often, there is a little of the next character type mixed in.