Owning a private key file when signing transactions allows to prove your identity to other parties. That is, what a blockchain uses to map identities to accounts/wallets, too.
Having a consensus protocol does not provide identity proving, but the eponymous censensus is based on a share of either state or computational power.
And state is just a database. State that decides what goes into the database is a recursion, voila an example would be Proof of Stake. Or as censensus protocols go, other names exist for the concept of proof of previously-done-arbitrary-things, which allowed you to mention a share with you cryptographic signature (i.e. hash derived from your private key) in the append-only database. Cases like a "plant a tree" eco-coins.
Also, to see the bigger picture on
> What distributed database would let you own your identity:
from a technically more complex angle: Tim Berners-Lee proposes a DRM system for users to own their content, instead of using this technology against user liberties. That would include owning your identity and enforcing your will on you intellectual property. I found these plans not feasible, yet, but at least people who know about cryptography still research the topics of user rights and P2P networking.
what distributed database would let you own your identity without any third party party being able to take it away from you?
that's a necessary condition for sufficient decentralization, and the only distributed databases capable of achieving such guarantees are L1 blockchains.