Matt Hardy and Jeff Hardy. Yes, they're brothers, but they're also a lot closer than most brothers their age would be...living within line of sight of each other, doing a webshow together, going out together, etc. They've built up a pretty substantial group of True Companions around themselves as well, with the rest of the OMEGA crew in close proximity.
And from TNA, we have the Motor City Machine Guns, who have used this exact phrase (specifically, Alex Shelley in regards to Chris Sabin) to describe each other.
Edge and Christianaren't brothers in real life, as they used to be billed, but they have been best friends since the sixth grade. They went to junior high together, went to high school together, went to college together, trained for wrestling together, broke into the indies together, went into the WWF together...Christian's TNA stint is the first time in their lives they weren't around each other constantly. The second being Edge's retirement.
Charlie Haas and Shelton Benjamin, aka The World's Greatest Tag Team, have a very strong bond with each other; in fact, Charlie has said in interviews that, sometimes, he thinks that his deceased brother Russ sent Shelton to him as a replacement.
The Miz and John Morrison definitely qualified during their time as a tag team. When they were separated in the WWE Draft, one announcer even stated "The bromance is over!".
Daffney Unger has called MsChif her better half (half of The Scream Queens, their tag team). MsChif has in turn said she can make Daffney's ASS whole. (Unger was scouting for another member of her All Star Squad).