I thought when I brought up the Wehrmacht mirror people would catch onto all the mirrors but apparently everyone's fixated on wehr. So let me bring up some other things...
Brit vs Brit
Can anyone imagine a more dull game. Mass emplacement spam because they suck against buildings. Brits also have very poor ways of dealing with trenches, so expect plenty of those. Arty would be the name of the game as you try to bring down the enemy city.
US vs US
The rifle battles would be unbelievably slow because they would always be fighting at max range because charging rifles is suicide, even when you use the same unit against them. The things that make US fun, the pace of the game and the brilliant flanks, would all disappear.
PE vs PE
Well, we all know how poor against early light vehicles PE are...
USSR vs USSR
Could actually be strangely fun. I'll admit I would want to try that.
The reason why mirror matches wouldn't work in CoH is because it is a game of situation, not like the standard RTS. It doesn't always matter what troops you have but how you use them.
Brit vs Brit
That would mean mass emplacements use but Churchill Croc and AVRE are made to kill emplacements.
Commandos can use nades vs trenches and Radio Triangulation to precise barrage repairing sappers.
Finnaly Priests! No arti is better than it Priests. It would destroy enemy emplacements very quickly.
Many people play bridge maps and shell one another with arti untill anihilation. UK vs UK would be a similar game.
US vs US
Rifles can be upgraded with nades, stickies and BARs so all infantry battles wouldnt look the same. A flamer squad is a cheap unit and it denies using green cover.
There are Thompsons SMG to fight at close range and WSC to make youre oponent move out of cover. WSC + Forward HQ (to produce rifles) would probably work well vs barracks+BARs.30cal and stickies can be used to kill M-8/T-17 so the teching up quickly wouldnt mean winning.
US vs US would be a fast game with a lot of micro.
PE vs PE
AT halftrack would finally be a good unit used for something more than treadbreaking. It would be used to counter AC spam.
AT nades and quick shrecks would be what infantry needs vs vechicles.
Panther battle group could be countered by AT infantry.
Everything that PE has can be counterer by PE.
88 vs Jagdpanther? I would love to see that.
PE would be a fast game. Quick reaction to build counter units would be needed.
Civil war games would be something else.
As youve said GodlikeDennis "It doesn't always matter what troops you have but how you use them".
If someone can hold a good defence but wont advance (cap) he wont tech-up. If someone attacks on all fronts he will loose steem quickly and wont be able to counterattack.
You need to attack weak points, defend when having less units.
If someone digs a trench and sits in it the whole game he wont win a game. Units need to be doing something, not just defending.