Looks like there is a bug in the Oracle Universal Installer 12.2, it does not register the ODP.NET components. You can run script below (modify folder names according to your system) in order to configure them properly.
set Oracle_x64=c:\oracle\product\12.1\Client_x64\odp.net
set OraProvCfg_x64=%Oracle_x64%\bin\4\OraProvCfg.exe
set Oracle_x86=c:\oracle\product\12.1\Client_x86\odp.net
set OraProvCfg_x86=%Oracle_x86%\bin\4\OraProvCfg.exe
set TNS_ADMIN=C:\oracle\network\admin
IF EXIST "%Oracle_x64%\managed\common\Oracle.ManagedDataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\managed\common\Oracle.ManagedDataAccess.dll
IF EXIST "%Oracle_x64%\managed\PublisherPolicy\4\Policy.4.121.Oracle.ManagedDataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\managed\PublisherPolicy\4\Policy.4.121.Oracle.ManagedDataAccess.dll
IF EXIST "%Oracle_x64%\managed\PublisherPolicy\4\Policy.4.122.Oracle.ManagedDataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\managed\PublisherPolicy\4\Policy.4.122.Oracle.ManagedDataAccess.dll
IF EXIST "%Oracle_x64%\managed\common\Oracle.ManagedDataAccess.dll" "%OraProvCfg_x64%" /action:config /product:odpm /frameworkversion:v4.0.30319 /providerpath:%Oracle_x64%\managed\common\Oracle.ManagedDataAccess.dll /set:settings\TNS_ADMIN:%TNS_ADMIN%
IF EXIST "%Oracle_x86%\managed\common\Oracle.ManagedDataAccess.dll" "%OraProvCfg_x86%" /action:config /product:odpm /frameworkversion:v4.0.30319 /providerpath:%Oracle_x86%\managed\common\Oracle.ManagedDataAccess.dll /set:settings\TNS_ADMIN:%TNS_ADMIN%
rem Oracle.ManagedDataAccessDTC was used in earlier versions, not used anymore
"%OraProvCfg_x64%" /action:ungac /providerpath:Oracle.ManagedDataAccessDTC
IF EXIST "%Oracle_x64%\bin\2.x\Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\bin\2.x\Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\2.x\Policy.2.102.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\2.x\Policy.2.102.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\2.x\Policy.2.111.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\2.x\Policy.2.111.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\2.x\Policy.2.112.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\2.x\Policy.2.112.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\2.x\Policy.2.121.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\2.x\Policy.2.121.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\2.x\Policy.2.122.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\2.x\Policy.2.122.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\bin\4\Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\bin\4\Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\4\Policy.4.112.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\4\Policy.4.112.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\4\Policy.4.121.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\4\Policy.4.121.Oracle.DataAccess.dll
IF EXIST "%Oracle_x64%\PublisherPolicy\4\Policy.4.122.Oracle.DataAccess.dll" "%OraProvCfg_x64%" /action:gac /providerpath:%Oracle_x64%\PublisherPolicy\4\Policy.4.122.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\bin\2.x\Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\bin\2.x\Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\2.x\Policy.2.102.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\2.x\Policy.2.102.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\2.x\Policy.2.111.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\2.x\Policy.2.111.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\2.x\Policy.2.112.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\2.x\Policy.2.112.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\2.x\Policy.2.121.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\2.x\Policy.2.121.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\2.x\Policy.2.122.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\2.x\Policy.2.122.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\bin\4\Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\bin\4\Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\4\Policy.4.112.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\4\Policy.4.112.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\4\Policy.4.121.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\4\Policy.4.121.Oracle.DataAccess.dll
IF EXIST "%Oracle_x86%\PublisherPolicy\4\Policy.4.122.Oracle.DataAccess.dll" "%OraProvCfg_x86%" /action:gac /providerpath:%Oracle_x86%\PublisherPolicy\4\Policy.4.122.Oracle.DataAccess.dll
Update
According Oracle Doc ID 2272241.1 this seems to be by intention:
Applies to:
Oracle Data Provider for .NET - Version 12.1.0.2 and later
Microsoft Windows x64 (64-bit)
Microsoft Windows (32-bit)
Symptoms
When installing the 12.2 Oracle Data Provider for .Net, the provider
is not automatically registered in the Global Assembly Cache (GAC).
Changes
Moved to the 12.2 version of the Oracle Provider for .Net
Cause
This is the expected behavior. Since so many environments are using
multiple versions of ODP.Net side by side, starting with 12.2, the
installer does not automatically register the ODP.Net provider in the
GAC by default.
Solution
Manually register the dll into the GAC as outlined in Doc ID
1169890.1: HOWTO: Example use of OraProvCfg.exe to add/remove Oracle assemblies from the Global Assembly Cache
However, I would consider this as a bug. It is quite difficult to run multiple versions of ODP.Net side by side, because the version of Oracle.DataAccess.dll
has to match exactly the version of the Oracle Client. Installing more than one Oracle Client (i.e. more than one each for 64-bit and 32-bit) typically causes a lot of problems and is rather uncommon.