FS's minimalist worldbuilding doesn't give much info about the structure of government within GTVA territory - we can easily conceive of multiple civilian governments of various colonies, each with different internal structures and cultures and attitudes, each a signatory to the GTVA. FS certainly doesn't preclude things like civilian police forces (a la Destiny of Peace's first mission), militarized fascist police forces, private security/mercenary corps, even potentially local militaries or militias that answer directly to a colony's civilian government rather than the GTVA itself. (Some of these are likely a point of tension between various colonies and GTVA leadership.)
Based on the very short FS timeline, colonization happened extremely quickly both before and after FS1, so there are reasons for all the colonies to consider themselves part of a single collective in some sense (including a common enemy: first the Terrans/Vasudans, then the Shivans). The GTVA's single-military-government structure seems like a relic of this time. Post-FS2, the culture is probably trending away from the desire for a single government, as various colonies start to establish their own cultural identities and want to manage their own defense forces. The GTVA could end up transitioning to a more NATO-style alliance of militaries rather than a single military entity unto itself, or its leadership might resist this change.
Keep in mind that there really is no "Russia" to the GTVA's "NATO". There are only two major political factions in known space and they are both part of the GTVA, so other than quelling the occasional violent rebellion, the GTVA's singular purpose as an organization is "Prepare for future Shivan invasion". Given how shortsighted people can be, I could easily see various colony governments questioning the amount of resources allocated by the GTVA for R&D and military construction that could be diverted towards welfare/civilian development.