How exactly do you start the game?
In my experience, the multiplayer and network option work only if the game is started via Galaxy, but not when directly starting NMS.exe (which is the 'NMS' program entry in my screenshot).

If you want Galaxy to autostart No Man's Sky, you can add the launch options:
/command=runGame /gameId=1446213994

I have the same issue that multiplayer only works with the bottles flatpak version but not in the bottles system (AUR) package. The downside of the flatpak version is that each game start is slow, it seems that shader caches (or whatever else is processed there) are not persistent.
I get better performance when running NMS with proton-ge via heroic, but no multiplayer/online service and Galaxy tends to crash a lot. Galaxy is more stable with heroic & wine, but NMS performance is so bad to make it unplayable.