Yes — Oxford’s electoral services privacy notice says the Cabinet Office and DWP use registration data to verify the identity of new applicants, and the Council compiles and manages the Electoral Register. If you want to see whether you’re on the register, the public-facing option shown in Oxford’s records is the Open Register opt-out / electoral services contact via elections@oxford.gov.uk.
If you meant a different kind of register, Oxford also has public registers for housing, taxis, and self-build, but there isn’t a page here that lets you directly check your own electoral registration status online.