You are using on both systems OCPN 5.0 and oeSENC 3.0?
And yes, you are right, the second system needs its own individual chart set.
When you follow the steps for the online install from the oeSENC tab of Options|Charts --> oeSENC, when refreshing the chart list and highlighting the set you want o install, you are getting prompted to select either an existing system or define a new one.
Has this been the case?
Don't try to install the set from the RasPi on the PC. This will not work
Under Options|Plug-ins|oeSENC --> preferences, which system name does show up there?
Btw.: the screen
shot in your post is not visible for me.