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 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).
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.
Lots of non-human cute girls doing non-human cute things in a fantasy world with magic and not much of a real plot. If you want something fluffy and bright with nothing much going on, these are two great candidates. Dog Days has a bit more 'substance' with the whole battle thing going on (although it's not serious in the slightest), but if you liked it, you may take interest in Zettai Bouei Leviathan.
Plenty of non-human girls running around in a fairly dull fantasy world, doing cute thing and generally acting pretty girly. Both contain a pretty boring plot with little to nothing at stake in the grand scheme of things.
Dog Days perhaps has a little more to it with it war games premise, but it's pretty marginal as both have a distinct lack of, well, anything going on most of the time.
Set in a fantasy world, both series follow a female cast which is often subjected to the usual ecchi tropes for anime of this type.
Each has a fairly shakey plot that includes several nations in the midst of conflict, as well as having plenty of 'fun' moments with girls acting all cute and girly. Neither are particularly original, but fans of one should give the other one a try as they share many similarities.
After suffering a heart-wrenching rejection, anime and manga-loving Shinichi Kanou shuts himself off from the world and becomes a hikikomori. But the boy's solitary life soon comes to an end when he's hired as an ambassador to a newly-discovered world, where the only job criteria is to be a hardcore otaku! Now the boy must work hard to spread the word of this unique, nerdy culture in hopes of strengthening relations between the other world's citizens and Japan. And in a land filled with dragons, half-elf maids, and a teenage queen, it shouldn't be that hard... right?
Each of these shows has a pretty similar premise, with each revolving around a main character who is summoned to another world and given the task of improving the relationship between two worlds, as well as helping to improve the standards of living along the way. There are also parallels in the way conflicts are handled.
Outbreak Company is a lot funnier, and generally an all-round better show than Dog Days, but anyone who enjoyed the themes of one should give the other a go.