The canon relationships I talk about here are platonic with a feeling of friendship or brotherly love. The relationships displayed here are the ones on-going in the manga. There is no romance between these characters whatsoever.
Tamaki & KyoyaTamaki and Kyoya are good friends in the manga, though it doesn't often show. Kyoya is often either ignoring or going along with Tamaki's dramatic scenes. He does not protest being called Mom while Tamaki is the Dad. Where Tamaki is busy handling the club from the front, charming customers and whatnot, Kyoya works from the background with his calculations and data. In Volume 6, they share several moments. Tamaki offers to give Kyoya a shoulder massage before the race for the Central Salon because he seems to be under a lot of pressure. Though Tamaki doesn't seem to know it, Kyoya admires him at times and believes in what he can do. In the most recent Ouran arc, Kyoya goes on the 2nd year school trip to France while Tamaki stays behind. Kyoya makes it his mission to find Tamaki's mother and bring back news. While Kyoya is often exasperated and/or frustrated with Tamaki, there is no doubt that they are best friends underneath it all.
Hikaru & KaoruHikaru and Kaoru are twins who share a close brotherly relationship. One is rarely seen without the other. They used to live in their own world of "us" versus "them". They still kind of do, but their world has opened up a bit since middle school and joining the Host Club in high school. They're not quite the same, but still share some common personality aspects. Both of them like to be daring and are usually the trouble-making ones. If there was a prank or some other type of mischief, they would be the first suspects. They have a care-free and easygoing attitude about a lot of things. Seriousness is a rare facet with these two. Though they are laid-back, both twins are clever and witty. They like to tease and flirt with girls. They play up their relationship to extreme heights, going so far as to suggest twincest, which may or not be their true relationship. (You can never tell with these two.) They also like to keep up their joking reputation. They only reveal their vulnerabilities to each other. In Chapters 50-53 (V11-12), the twins go through a serious phase that threatens their entire relationship. However, in the end, Hikaru steps up as the big brother and all is right with the world.
Hunny & MoriHunny and Mori share a close friendship, especially being cousins and the only third-years within the Host Club. As we learn in Volume 2, they are cousins and childhood friends. The Morinozuka family served the Haninozuka family for generations before marriage blended the family two generations ago. Still, Mori sticks close to Hunny. Hunny can usually be seen in Mori's arms or getting a piggy-back ride from him. When Hunny cries and gets upset, Mori is the first person he runs to. Meanwhile, Hunny waits for Mori when the older boy has Kendo practice. In the extra episode titled "Hunny's Bun-Bun," Mori is the only one who knows how to handle Hunny, while everyone else is freaking out about what to do with the stained Bun-Bun. Opposites seem to attract in this relationship because where Mori is quiet and sober, Hunny is outgoing and cute. However, they seem to suit each other just fine and their unique dynamic only leaves the reader with one word: "Awww..."