Ok, after about 2 hours of troubleshooting with a friend we may have found out what was going on, but have no idea why this would be causing the issue in the first place. Trying to recreate the bug under the exact same conditions proved pointless. Whatever caused the bug to appear in the first place could not be reproduced by making new games and playing them to the same point in time and ensuring autosaves matched (loading an old game with no autosaves/both players having the same or no autosaves for the same game). However, we were able to find a solution that seems to have worked to get the bugged lobby to function, but it does not fix the bug if we reload the same lobby again. 

   The way we got to the lobby to work was composed of two parts as they seem to both be connected to the issue. When loading into the lobby as the host I saw the same "Name of Competitive Faction" text for the eight competitor slot with no faction selected as is shown in the earlier screenshot I posted here (different game, but the same problem). I tried joining that eight competitor in the game on my own to see what it was and loaded into the game. As the screenshot shows it was a Lumeris AI. The game did not load in, but I now had the correct faction for the eight bugged competitor slot. After quitting and loading back into the lobby after a fresh restart I changed the eight competitor to Lumeris and then tried to load into the game and it worked correctly for the first time. I then tried to see if my friend could join and play the game with the correct AI selected and the lobby was still bugged for us. My friend noted that the AI list for him was different than what I ended up seeing. He saw the eight competitor slot, which I had determined was Lumeris, as instead being his custom faction (which he was not playing by the way) as being the actual eighth competitor. We had the idea to delete this custom faction of his and try the lobby again after a game restart. Coincidentally, after setting the eighth competitor to Lumeries again, when my friend joined again we both saw the same list of competitors. My friend was able to ready up and we were both able to load into the game! We tried this again without setting the eighth competitor correctly and the lobby bugged out again and we were not able to load into the game so we confirmed that both of these conditions needed to be met in order for the game to work. 

   Like I said before I have no idea what seemingly caused the game to forget what the eight competitor was for the game, but it could have been the custom faction my friend had all along or maybe not. Maybe it only came into issue on reloading the game and not at any other time. I hope that anyone else that runs into this issue is able to have similar success and that this problem does not get buried to the point where newer players who encounter this awful bug just decide to not play anymore as this bug is quite game breaking for multiplayer to say the least.

Not sure so far, but perhaps the issue has the same source as posted in this thread?

https://www.games2gether.com/amplitude-studios/endless-space-2/forums/73-tech-support/threads/34909-error-occures-when-i-m-starting-a-new-game-patch-v1-5-8-s5-gold-public-steam