"Functional Database" is contradictory. Honestly this sounds like a bunch of buzzwords concatenated together. Is there an example or a purpose for this?

Personally, I've heard of it in the context of https://github.com/commanded/commanded, a CQRS framework for Elixir that supports EventStore as a storage backend.

I think the idea is that it's something simpler than a database. It's more like an append-only file that has on-insert triggers/durable running queries. (Sort of like https://www.pipelinedb.com/ does, or like blockchain nodes do.)

Or, you could think of it as message queue like Kafka, with permanent durability of all "messages", and a single fixed subscriber bolted onto the queue server, where that subscriber is exposed through the queue server's API, allowing users to reprogram it arbitrarily.