A finite factored set is "just" a set with a specific choice of decomposition as a product of sets . I'm not sure what definition of phase space you're using, but for a sufficiently general definition of dynamical system (e.g. https://en.wikipedia.org/wiki/Dynamical_system#Formal_definition) I don't think that the phase space necessarily has coordinates in this way. The position / momentum phase space example is a special case, where your phase space happens to look like a product of copies of the real numbers, which is then getting back to the "factored" part of a finite factored set. I'm not convinced that there's a deep connection here, though am not very familiar with either concept so could easily be missing something here!
Garrabrant's finite factored sets feel to me like the same thing as a phase space of a dynamical system. The differences I can see are that phase spaces are not always finite, and that finite factored sets don't have the context of a dynamical rule defined on them. They seem to share the property that every element in the set has exactly one coordinate in each dimension/is an element of exactly one partition of each factor.