Though this does not answer how to put any current database you have right now under version control, in the last few years some excellent databases have been created with the aim of making the database work like git, meaning all your data is branchable, time-travellable and auditable. Personally, I think TerminusDB is the most promising one.
- Irmin (branching + time travel)
- Flur.ee (immutable + time travel + graph query)
- XTDB (formerly called 'CruxDB') (time travel + query)
- TerminusDB (immutable + branching + time travel + Graph Query!)
- DoltDB (branching + time-travel + SQL query)
- Quadrable (branching + remote state verification)
- EdgeDB (no real time travel, but migrations derived by the compiler after schema changes)
- Migra (diffing for Postgres schemas/data. Auto-generate migration scripts, auto-sync db state)
- ImmuDB (immutable + time-travel)
Honorable mentions
- Neon: Hosted serverless postgress (branching)