I would say identity is core to objectness. Without identity, something is purely structural, only its value is relevant because it will not transition to another state later. Objects must be things you can manipulate. They have a presence beyond their values.
I use objects that can't transition from one state to another practically every working day. It's a strange view point to me that because something is immutable, it's not really an object. But a view worth thinking about.
To me the important thing is hiding internal state and sending high level messages instead of something like "change internal field x to y". Whether that internal state you're hiding is immutable or not seems irrelevant to me.
If you have a variable X that is pointing at different immutable maps over time, then X is the identity of your object that is implemented with multiple immutable maps.
