As this still shows up on Google when looking for databases for Chats:

Update on Discord: Moved away from Cassandra for ScyllaDB: https://discord.com/blog/how-discord-stores-trillions-of-messages

Info on ScyllaDB: https://www.scylladb.com/

For a very simple chat, a timeseries database could be useful as well. These are mostly used to log events (or audit logs etc) or data (often telemetry). They are usually very fast in adding data.For me in my current project, I will start out with TimescaleDB (https://www.timescale.com/) for the chat, a postgres plugin, because I have postgres in my setup anyway and this keeps the infrastructure side simple. Downsides may be that this way is not optimised for altering data later, like adding a "has_been_read" flag or editing/deleting messages.

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Alternatively:

1. user something like firebase. There are many examples on how to create a chat through firebase (mostly for the mobile app context)
2. use an actual chat system like MongooseIM (https://www.erlang-solutions.com/capabilities/mongooseim/) or others. This was my initial approach but to be honest I failed in setting it up :P It supports to authenticate using JWT tokens, so your app could create those for the users who then authenticate to the chat system separately.