A player's country (or army) in AWBW is essentially their chosen faction for a game, and is largely just a cosmetic change. Aside from affecting the colour of owned units, it will also affect the sprite appearance of Infantry, Mechs and the HQ and whether their units face left or right. In AWBW there are 18 available countries to choose from, in contrast with the main series' which features only up to 4 (or 5, where Black Hole is available). A further 2 countries are on the way pending the determination of their appearance.
The player's country may be changed both before the game starts (which will be the default country choice for other players when they load in) as well as in-game by clicking on the small drop-down menu next to each player's name. Especially for colourblind players, it may be prudent to select a country with a colour that is easy to distinguish from the opponents', and ideally faces the opposite direction to them for further differentiation.
In Design Maps, the default countries chosen on a map has more significance as they determine the move order for each player, as players with a country higher in the move order will always move before those with a lower order one, even if a player elects to change to a different country before a game starts. The chart below shows the countries in AWBW and their respective turn order.
More info on the countries' units, themes, and random trivia are available here.