Regarding "using shared": that's a wrong and confusing hint in this case. If you have a multiple item/characterScripts that depend on the same scripting via e.g. "USING Base" and then assign these scripts to the same item/character, you will get errors about the fact that all global variables of Base are now created twice for that one item/character. By changing the "USING Base" to "USING SHARED Base", you can tell the behaviour script system to instead share the copies of those global variables with other scripts. The script compiler gives a hint about this when it detects duplicate global variables.
In case of a game script, you don't assign them to characters or items, but to mods. Since they don't belong to an individual object, all of their variables are global (including the ones locally defined in events). So if you have multiple mods with gamescripts that define the same events with the same variables, the hint kicks in like before. However, in this case it does not make sense and you cannot fix anything this way.
Back to your original question, I made a mistake. You can override Osiris/Story scripts by including a script with the same name in your mod. However, you
cannot do this with item/character/game scripts. The reason is that these are not looked for based on their name, but based on the UUID of their
resource. So you are now including both the original game script and your own, which is causing the conflicts (and which is not what you want).
UUIDs are how most objects in the game are resolved, such as characters and items. The way to override these, is to include an object of the same type with the same UUID as the original one. That's what the editor does when you modify e.g. a character from the DivinityOrigins mod in your own mod: it creates a copy of the original character with the same UUID in your mod, and then stores your changes in this copy.
In theory, you should be able to do the same with the resource of the game script: go to the resource manager, find the resource for the game script you want to override, edit it, and set your own game script as the script it should use. However, I've heard this is currently not working. They are looking into why that is the case. Sorry I don't have better news.