- Irmin (branching + time travel)
- Flur.ee (immutable + time travel)
- Crux DB (time travel)
- TerminusDB (branching + time travel)
- DoltDB (branching + time-travel)
I have been looking for the same feature for Postgres (or SQL databases in general) for a while, but I found no tools to be suitable (simple and intuitive) enough. This is probably due to the binary nature of how data is stored. Klonio sounds ideal but looks dead. Noms DB looks interesting (and alive). Also take a look at Irmin (OCaml-based with Git-properties).
Though this doesn't answer the question in that it would work with Postgres, check out the Flur.ee database. It has a "time-travel" feature that allows you to query the data from an arbitrary point in time. I'm guessing it should be able to work with a "branching" model.
This database was recently being developed for blockchain-purposes. Due to the nature of blockchains, the data needs to be recorded in increments, which is exactly how git works. They are targeting an open-source release in Q2 2019.
Because each Fluree database is a blockchain, it stores the entire history of every transaction performed. This is part of how a blockchain ensures that information is immutable and secure.
Update: Also check out the Crux database, which can query across the time dimension of inserts, which you could see as 'versions'. Crux seems to be an open-source implementation of the highly appraised Datomic. However, it doesn't support branching like Dolt and Git.
Crux is a bitemporal database that stores transaction time and valid time histories. While a [uni]temporal database enables "time travel" querying through the transactional sequence of database states from the moment of database creation to its current state, Crux also provides "time travel" querying for a discrete valid time axis without unnecessary design complexity or performance impact. This means a Crux user can populate the database with past and future information regardless of the order in which the information arrives, and make corrections to past recordings to build an ever-improving temporal model of a given domain.
Update II Check out Terminus DB: "Documentation for TerminusDB - an open-source graph database that stores data like git". It supports branching like Dolt and Git, but does not look very mature yet.
Update III NomsDB was forked and is now in use in DoltDB: https://github.com/dolthub/dolt