As far as I know, the connections selection can just show the connections in the same resource group. So you need to put the logic apps in the same resource group.
In my side I'm not able to test "File system", I test the blob storage connection in my visual studio(I think they are the same thing). I create the first logic app in visual studio and right click the "LogicApp.json" and click "Open With Logic App Designer". I choose a resource group in the popup window.
![enter image description here](https://i.stack.imgur.com/imbNv.png)
Then I use a "Get blob content" action in the logic app in visual studio. It create a blob storage connection.
![enter image description here](https://i.stack.imgur.com/gDYey.png)
We can also see this connection in the resource group in azure portal.
![enter image description here](https://i.stack.imgur.com/ytzyJ.png)
Then I create second logic app in visual studio and when click "Open With Logic App Designer", I choose the same resource group as the first one above. In this second logic app, when I use the "Get blob content" action, it can connect by the same connection which already exist in the resource group (The screenshot below comes from the second logic app in visual studio although it is same with the screenshot above).
![enter image description here](https://i.stack.imgur.com/lcYVO.png)
If your logic apps are not in the same resource group, I don't think we can use the same connection. And by the way, I think edit the logic app in azure portal directly is easy and convenient. So I would suggest you edit your logic app on azure portal rather than in visual studio.
Hope it helps~