Deku's Battle Cry isn't approved by the other "big-named" Anime heroes makes sense on a meta-contextual level of his character. The ones reviewing his performance are Goku, Vegeta, Naruto, and Ichigo, and all of them are naturally Hot-Blooded as part of their Stock Shōnen Hero archetype (yes, even Vegeta as he's an Unbuilt Trope of the Stock Shōnen Rival). Izuku, on the other hand, is an analytical Shrinking Violet, making him the opposite of the usual stereotype associated with the Shonen hero.
Boruto being a No-Respect Guy among the newer Anime characters makes sense due to the Broken Base regarding his Spin-Offspring being either better or worse than his father's series.
Why are the members of the Cartoon House's vigilante group: Robin, Aang, Ben Tennyson, and Samurai Jack? Well, each of them in their own respective series dealt with main enemies hell-bent on world domination and oppression (e.g. Slade, Fire Lord Ozai, Vilgax, and Aku), making their grudge against the Anime House's discrimination and segregated status quo more understandable.
On a deeper level, the conflict between Cartoons and Anime makes sense due to the Values Dissonance between American and Japanese cultures. note In America's individualistic culture, rebelling against authority is seen as a good thing, or at least, if said authority is too oppressive (do note that such viewpoint can vary upon each state). Japan on the other hand, along with most of the ones in eastern Asia, is a collectivist culture that heavily frowns on their citizens not putting the group harmony first regardless of any justifications, speaking ill of any political leader is blasphemous (Either you speak praise about him or her, or you just stay silent, as you are expected to have absolute loyalty to your country and prioritize their beliefs over your own. In more extreme cases, you pretend you like them.), and going against the established status quo is never seen as a good thing. Being the product of Japanese-created media, the Anime House has no problems with either enforcing the status quo or standing on the sidelines as the Cartoon House is suffering from being segregated and discriminated against, and in turn, the Cartoon characters (all of them being a product of American media franchises, even the Animesque ones) rally together to form a vigilante group to fight against the oppression.
While Frieza's Curb-Stomp Battle against New Era Naruto Uzumaki through a casual Death Beam seems like a serious case of Power Creep, Power SeepAss Pull, it should be noted that he already had his "Black Frieza" form from the Super manga and that Super Mode allows him to casually curb stomp both True Ultra Instinct Goku and Ultra Ego Vegeta with a strike to each Saiyans' abdomens.
Anime House the Animated Series
Beast Boy's anger towards the discrimination and segregation that the Anime characters have towards the Cartoon characters makes sense due to his Dark and Troubled Past in the comics.
Compared to Beast Boy, why is Danny Phantom unwilling to tackle the Anime House's discrimination and segregation of the Cartoon House? Well, besides the Anime characters' power and skill are on a "whole 'nother level" compared to the Cartoon characters, it's likely that Danny speaks from experience in his own show; Danny's high school bully, Dash Baxter, would often get away with his bullying of Danny and his friendsdue to him being the school's star football player (though Danny would sometimes get back at Dash by using his ghost powers in subversive ways and it's foreseen that Dash's future will plummet once he graduates if he doesn't change his ways) and throughout most of the series, Dash doesn't let up of being a Jerkass despite Danny helping him out at times till the Series Finale after seeing Danny saved the whole world.
This means that Danny knows that Beast Boy's attempts to gain respect from the Anime House by simply going out to assault, battery, and cause collateral damage won'twork due to the cards already stacked against the Cartoon characters through the discriminatory and segregation policies and the "dim view" they already have of them. A good example would be seen when Danny prevents Beast Boy from taking a swing at Rin and Mustang. While Rin and Mustang were very much acting like smug jerks and barring them entry to the Crossover Festival, along with the implication that Seto Kaiba uses Very False Advertising, they still have done nothing to warrant such retaliation. Not only would they give a No-Holds-Barred Beatdown in "self-defense" if Beast Boy succeeded in punching one of them, along with the implication that they would kill him, Danny, and Jack and can even frame their deaths as involuntary manslaughter if criminal-intervention become inevitable, but it would only confirm all the bad things that the other Anime characters have said and thought about them.
Furthermore, much like Real Life bigots, sometimes, they won't let go of their prejudiced views no matter what. Implying that while it's possible for the Cartoon House to convince some Anime characters that they're worthy of respect, others would not be so accommodating as seen with Zeke and Eren, Kaiba, Mustang, Rin, and probably Bakugou.
While each character is animated with their own respective series' art style, the general one for the series as a whole is from My Hero Academia. Given the overall themes of My Hero Academia and the Cartoon House's motivations for storming the Anime House, it makes sense.
