Yes, you can select what the colors in the Galaxy map should indicate (Government type, allegiance, etc.).
Each system has one principal allegiance. That is the one shown in the Galaxy map. However, this does not mean that there are no other factions active in that system. You can look at the factions active in the current system by checking the right-hand panel. You should always do this before accepting any missions so you know which of the system factions are aligned with the Federation - there is often more than one faction in a system that is aligned with the Feds. To build up reputation with the Feds, choose ONLY missions from those Fed-aligned factions. (You may choose missions from other factions too, but sometimes this will lead to rep loss with the Feds.)
The fact that there are always several factions in one system also means that you can find Fed-aligned factions in a system that does not show as Federal in the Galaxy map. Whether a system is shown as Federal in the map depends on whether the main station in a system is controlled by a Fed faction or not.