Maybe I've been spending too much time with the state monad, but that still sounds like a function to me. The fact that the DB contents get passed in implicitly as part of the context of a Rails app does not change that.
There's a difference between a "function" that uses implicit contextual state and one that doesn't. We can call it a function with implicit context or we can call it an object or a service or a process. It doesn't really matter as long as we have some way of talking about it. Those other words have been in common use for a long time so it's probably more convenient to use them.