Welcome to the Little Garden, where people band together in like-minded clans, parallel realities intersect, and the truth of stories comes to life! The masters of this world host the Gift Games, in which extraordinary prizes may be won and lost by gods, spirits, monsters, and people of extraordinary ability. One dwindling clan has lost even its own name in the Games, and seeks new talent to make a comeback. Black Rabbit, a games referee, has brought Izayoi Sakamaki, Asuka Kudou, and You Kasukabe, three individuals with unique abilities, to this world to help return this clan to its former status. For the struggling clan, the stakes have never been higher; but are the newcomers just out to make a name for themselves, or will they really help the clan succeed?
In the magical land of Flonyard, animal-eared inhabitants battle each other for resources in harmless war games. One region, Biscotti, has fought valiantly but is on the verge of financial ruin when its princess, Millhiore, makes a decision: she’ll summon a champion who can fight for their honor! Through a magical portal comes Shinku, a once-typical middle school student who’s eager to join the cause. But soon, both Millhiore and Shinku make a terrible discovery: those who are summoned can’t return home! Now, as Shinku continues to train and fight for Biscotti, he and the others must also try to find a way for the boy to someday get back to his loving friends and family.
Mondaiji-tachi ga Isekai kara Kuru Sou Desu yo? is more serious and less fluffy than Dog Days, but they both include characters being called to a fantasy-like world by a female with animal qualities. The summoned "heroes" are then asked to aid the summoner with protecting their community (Mondaiji-tachi) or their country (Dog Days). In order to do this, the heroes must fight in a non-traditional way; through games/competitions.
The feel and art style of these two anime are almost identical. Very bright colors, pretty art, animal-human-crossbreed characters and different groups "fighting" against each other via magical games. Both are fairly light-hearted and a bit silly, however Problem Children is DEFINITELY the more serious of the two. Fans of one of these will almost definitely find the same enjoyment in the other.
Both animes are about the protagonist(s) being thrown into a mysterious world to help out what would be called a kingdom. In Dog Days, the protagonist, who is physically fit, helps out the kingdom with comedy involved, while in Mondai-Ji, the male protagonist and two female protagonists, who have certain powers, help out the kingdom using epic, action-packed warfare!
While Mondaiji-tachi ga Isekai kara Kuru Sou Desu yo? is noticeably more serious for the most part, both anime feature a cast who are taken to a strange fantasy style world by a female with animal characteristics. These characters are then asked to do battle in a series of high stakes games at the request of the female summoner.
Mondaiji-tachi ga Isekai kara Kuru Sou Desu yo? is also funnier, and has more than a few nods to mythology. But the concept is much the same in both.
Both animes have a main character that is super powerful, arrogant, smart, and was summoned to another world for one reason or another. Both animes have some form of fanservice... Hagure yuusha is ecchi very heavy.. Problem Children is more toned down and acceptable.. But if you like one your pretty much guarenteed to like the other
Both animes feature super-strong and talented protagonists that have their lives changed after visiting another world versed in magic and fantasy. The action is similar between both, though Estetica is a lot heavier with ecchi elements.
After inadvertently managing to kill the god Verethragna, taking his divine powers in the process, the once-typical teen Godou's life changed forever. Now a "Campione", a god slayer, the boy constantly attracts both gods' and other Campiones' attentions alike, not to mention the affections of a powerful knight, Erica. Whether he's dodging Erica's advances, proving his strength to others or taking down gods who will wreak havoc on the planet, Godou's new life is anything but normal.
Both of these series feature flashy action and a main character who's quick on the uptake as well as a harem of girls who fight by their side. These two animes also intertwine themes of fantasy and mythology that accentuate their respective plots.
In both anime, three humans are summoned to another world that first appears to be nothing but fun, until they're pulled into a fight to save it. They both include colorful fight scenes, a pretty good storyline (probably not the best, but definitely not horrendous), animal eared residents of the alternate worlds, and some fanservice (I probably wouldn't call either of then ecchi though).
Lucy is a seventeen-year-old mage with the power to summon stellar spirits, but what she really wants to do is join a guild - and not just any guild. She has her eyes set on Fairy Tail, a notoriously reckless and outrageous group of magic users who are likely to be drunk or destroying buildings and towns in the process of completing a job! While in town one day Lucy meets a perpetually-seasick boy named Natsu who, through a series of events, reveals to her that he's none other than the fire-eating mage Salamander of Fairy Tail! Lucy is finally able to join Fairy Tail and quickly begins to take on odd jobs with Natsu and his gang for fame and profit. Along with her fiery friend, Happy the flying cat, Natsu's archrival Gray and their overseer the invincible and beautiful Erza, Lucy sets forth for epic adventures that leave an epic amount of destruction in their wake. After all, destroying a city or five doesn't matter as long as they get the job done... right?!
Both of these anime involve an organized group of people with magical powers of some kind (communities in Problem Children and guilds in Fairy Tail) competing against other groups. Thanks to the colorful casts of characters, varied powers and comradare between guild/community members fans of one will probably find some enjoyment in the other, despite the huge disparity in series lengths.