I started a game from an installation I had, but then the game updated. I restarted the game within GOG Galaxy and while the save is still there, I can no longer scan anything. Also, those annoying little sentry robots no longer appear which is messing up my quest lines. I am considering just uninstalling the entire game and trying over. Why does starting the game from two different icons start two different versions of the game using the same saves? This is causing a lot of incompatibility issues.

I am not experiencing any problems with scanning, or sentinels, with NMS 2.61 installed.

Not quite sure what you mean about two different icons and different versions of the game. If you installed the game with GOG Galaxy, and run the game through GOG Galaxy, GOG Galaxy will keep the game up to date - Though it can update the game in the background, so next time you run the game any new version installed while you were playing will be the version you play next time you start the game (depends on the settings you have in GOG Galaxy).

If you installed the game with the Offline Manual Installers, that will not auto-update, and only updates if you grab more up to date Offline Installers and install over the previous version.

I guess if you use both methods you could have two different versions of the game installed on different parts of your hard drive. Depends if you chose a custom installation path to install one or the other.

Save games are installed in one location, and whatever version of the game you are using the same saves will always be used. Usually in C:\Users\<username>\appdata\roaming\hellogames\nms\defaultuser\

Incompatability issues : I can only guess you mean with Mods, a lot of mods need updating to be compatible with the latest version of the game, if they are old mods you may be using files taken from an older version of the game by the mod author, which can revert changes to the game made by Hellogames .. Causing issues. Even if the mod does not seem related to the issue, they can still be the cause. Maybe the source of your Scanning and Sentinel issues?

Edit: Note also that periodically Hellogames force a reset on whether you have a file called DISABLEMODS.TXT in No Man's Sky\Gamedata\PCBanks\ - This tells the game to ignore any mods installed. They do that to ensure the game runs if you have potentially game breaking mods installed which may have not been updated, its kind of like a flag to get you to check if any mods you have installed need to be updated before you continue playing.
I dont use any mods so cant tell you what the score is with whatever you have installed, but I do know the latest update v2.61 of the game did force a reset on that file.