ArcMap is a 32-bit application and you need to install the 32-bit version of Oracle client drivers like Oracle Instant client.
Then you need to add both the 64-bit and 32-bit oracle clients in your system path. 64-bit path first and followed by the 32-bit Oracle bin in the path.
You need to re-start ArcMap before attempting to connect so it can pick up the updated path variable.
I can work with you on this in a meeting.
You can contact me at jgough@esri.com.
Hope this helps.
Thank you,
Jim Gough.
Thank You,
Jim Gough | Technical Lead Enterprise - Educational Services
Esri