ArcGIS Desktop is a 32-Bit Application and hence needs the Oracle 32-Bit client. Whereas, ArcGIS Server is 64-Bit and hence will need the 64-Bit Oracle Client to be configured.
If ArcGIS Desktop and ArcGIS Server are on the same server, you need both the 32-bit and 64-bit clients. And in the Environment Path Variable, have the 64-Bit Oracle Client before the 32-Bit Client. Once done, restart the ArcGIS Server Service and then try Registering the Database.