Earlier quoted context omitted.
That is mutable state. Your Redux store is a mutable state container. If your redux store returns a {a:1,b:2}, and then you dispatch an action and it now returns an {a:2,b:1}, then you have mutated your state . It doesn't matter at all if the store's data structures are mutated in place or replaced with a modified immutable updated copy of a previous state, you still have mutable state .
I think I see where you're coming from, but I also think you might be moving the target a bit. I suspect you'd garner less debate if you said, I dunno, just "a dynamic UI app needs a way to model state changes ". "Mutate" has a specific meaning, in my mind at least, when discussing variables and data structures. React enables avoidance of direct DOM manipulation, Redux obviates direct state mutation.
If React lets you execute arbitrary JavaScript code (which it does), then you're no longer using just a finite state machine, you're using a Turing machine, which is something different, and more powerful. It may have many finite state machines embedded in it, but to refer to a user interface as a finite state machine is like referring to a car as a pulley, just because the engine requires several pulleys to work, and ignoring all the rest of the engine and car. You can't make a car out of just pulleys, and you can't make a user interface out of just finite state machines.