Can PDA model Turing Complete objects if the objects' state are finite?
where the domains of the object variables are restricted to finite sets. Hence, such objects can be modeled with a pushdown-automaton (PDA).


You are correct that if you understand it as using a pushdown-automaton to model the full behavior of the object, since as you indicated, you can put a Turing complete Turing machine inside any object.

However, I would believe that the authors intend to use a pushdown-automaton to mode an object in the domain with its possible states as defined by the values of its fields and its execution stack, since the whole paper is focusing on the ECF(Effectively Callback Free) property for objects. The actual moments when the object is still in the middle of changing from one state to another state are ignored/irrelevant/transparent/skipped. More specifically, the following shows the purpose of the object.


  Such a PDA for an object o is able to simulate any modular well-formed execution $\kappa \in \pi$ where the active object of all states in $\pi$ is o