Visual Studio is a 32-bit app. It won't have access to x64 drivers (like you said you installed) within the IDE itself. Of course your app (if it is x64) can use such drivers but the IDE won't see them. Try installing the x86 version of the drivers instead.
As an aside you mention you're getting an Oracle error. That isn't related to ODBC and I have no idea what this InstantClient tool is. IBM/Oracle provides drivers to talk to their databases. That is what you need to be using. To get them you need to download the appropriate drivers from Oracle directly. For a .NET app you can use either their legacy ODP.NET drivers or the newer Managed Data Access drivers that are available from NuGet. The NuGet approach is simpler as you don't need to install any drivers to get it to work. However it only works with newer Oracle versions.