The central idea is that each process is a functional program over an input stream of messages. The result from the functional program is an output stream of messages to others. From this perspective, Erlang is a rather clean functional language; there are no destructive updates to data structures (like setcar in Lisp and most Schemes).
With few exceptions, all built-in functions such as operations on ETS tables also follow this model: apart from efficiency issues, those BIFs could actually have been implemented with pure Erlang processes and message passing.
So yes, the Erlang language is functional, but a collection of interacting Erlang processes is a different thing. Each process is an ongoing computation, and as such it has a current state, which can change in relation to the other processes. Even a database is just another process in this respect.
In my mind, this is one of the most important things about Erlang: outside the process, there could be a storm raging, but inside, things are calm, letting you focus on what that process should do - and only that.
