Reference could not be added because of same name

Remove the reference to that dll and then reapply it. it worked for me.

The issue is that the reference name takes on the project name. It's still referencing the DLL by the appropriate name (A.Core.dll or B.Core.dll), but the reference uses the project name inside VS:

If you look at the project file after adding a reference to A.Core, you'll see this:

<ProjectReference Include="..\ClassLibrary1\Core.csproj">
  <Project>{16c42795-f703-4673-98e4-98758c09f9e0}</Project>
  <Name>Core</Name>
</ProjectReference>

Change the Name value to A.Core and you'll be unblocked. (B.Core will still show up as "Core" until you edit it as well.)