I'm sure that with everything going on right now about the the legal rights to the intellectual property and how Stardock plans to incorporate them, this may be totally reworked, but I do remember a lot of the original details that Paul Reiche wrote about the back story of Earth's introduction into the conflict.
If you revisit the original 1990 game manual for SC1, there is a narrative excerpt of Earth’s entry into the Alliance of Free Stars. Aside from that, there wasn’t a whole lot said about how Star Control operated independently on Earth before joining the Alliance. Paul Reiche did not go into abundant detail on it.
I always imagined it to be fundamentally the same idea as Starfleet in Star Trek, basically an interplanetary (and later interstellar) extension of the navy and air force. Similarly to Star Trek, the countries of Earth were united into a global government in the aftermath of a cataclysmic war. The “high provost” of the government was named Ivana Or-Kochov and was essentially the president or prime minister of Earth who signed the treaty to join the Alliance, although I guess she was technically separate from Star Control or any other armed forces.
Keep in mind that even Paul retconned some details of the canon between SC1 and SC2. For example, there were adjustments made to when the story took place. SC1 was originally set several centuries further in the future than SC2. Many background details of the story were told in an anecdotal format in the manuals and and it’s challenging to refer to a single accurate timeline between the two games.
I've a feeling that since SCO predates SC1 and it's a reboot anyway, all this canon is likely to be rebooted and/or reworked as well.