Is ODI Used by Oracle in their products?

Yes there are many Oracle products that utilise ODI, but here are just a few:

Oracle Application Integration Architecture (AIA)
Oracle Agile products
Oracle Hyperion Financial Management
Oracle Hyperion Planning
Oracle Fusion Governance, Risk & Compliance
Oracle Business Activity Monitoring
Oracle BI Applications also uses ODI as its core ETL tool in place of Informatica , but only for one release of OBIA and when using a certain source system.