Generally the issue of doubles is that some of their abilities scale *way* to well with themselves. The games I play in never allow doubles, but in cases where the opposing team presses the issue we allow it to demonstrate why we don't allow it, usually resulting in their most brutal death.
The other issue is that while doubles are indeed counterable, it generally requires you to either go doubles yourself or recompose your team to deal with the ability scaling of a double. However this just proves how damn well balanced this game is, each and every demigod possess one or more powerful character-defining ability.
A lot of the time the no double rule is for variety, though when I used to be able to play 5s, nobody cared if both sides had a double due to issue of finding 5 people who could play a unique demigod competently and there only being 8 whopping demigods.
For those who mentioned Sedna and Queen of Thorns, that combo isn't that much of an issue due to Queen's lack of burst damage and susceptibility to stuns. Double Queen isn't anything special either since Bramble Shield doesn't scale well past level ~7 or so since it doesn't take into consideration armor mitigation. Now if Bramble Shield could prevent debuffs while on...
I noticed people kept mentioning Snipe, Spit, Mines, Heal, but there are many other deadly abilities that scale incredibly well due to doubles.
Double Silence: Imagine two Sednas getting this, putting you into a full ten seconds of not being able to use any abilities mid-game.
Double Shield: At rank 3 or 4, you can't interrupt, stun, or slow a shielded character, 3000hp pots anyone? Good luck with the killings. Don't forget at max rank Shield prevents the Oak himself from being afflicted by debuffs when Shield is used.
Double Foul Grasp: I have never seen Foul Grasp trigger Stun Immunity(tm) but I could be wrong. Stay a while and listen!
Chain Boulder Roll + Hammer Slam: This hurts...a lot especially in larger matches.
*Mass Charm and the Aoe Freeze trigger stun immunity on creeps but I don't know if they trigger it on demigods.
*Double TB is still just as susceptable to stuns and silence so no scaling danger there.
I'm not fully against doubles, I just find it very boring to fight with or against. However that may be due to me getting very bored of this game.