Sub web.config does not override the parent regardless of the application. All web.configs will add up to the root of the main application. In order for the sub-application folder to use the existing connection string key, it must first be deleted or all connection strings must be cleared. If you want this to be a truly standalone application as a child, add it to your connection strings:
<connectionStrings>
<clear />
<your connection string>
<connectionStrings>
, :
<remove key="yourConnectionStringName" />