Actually, it was not really a multiple faction choice (factions are like USA vs. North Vietnam), but in WinSPMBT and WinSPWW2 it is possible to select Ally units under the Ally flag. They will be under player's control but they will belong to Ally. For example, Soviet Union can also use some Czech units under czech flag.
Is it possible to implement something like this later into the game?
It can be possible, but I think it would be unrealistic to have a HQ unit commanding platoons of various nationalities. I believe that in real life they would be operating inside their designated boundaries, in separate areas. Considering the dynamic nature of the game, it would be more realistic to play two smaller tactical battles (one with each faction) in a campaign, instead of one large battle with mixed factions.
The second question is about unit icons on the battlefield. In free version I tried to make some unique icons for special regiments (like color emblems) but all the color is destroyed by blue/green/red overlay. Is it possible to, somehow, remove this overlay and make possible to add colored emblems?
I suppose you're talking about the formation "flags". This is a good question. I think it should be possible to give them custom colors.