How can change in entropy be the same for all processes if the entropy production $\sigma$ is present for irreversible processes?
When a system changes from state 1 to state 2, the change in entropy of the system is the same for all processes, because entropy is a state function.
The difference between a reversible and an irreversible process is that, in the latter case, there is net entropy production for the universe, i.e., for the system+surroundings.
Since the entropy change for the system is independent of the process, the difference in entropy production between rev. and irrev. processes is seen in the entropy change of the surroundings.
More broadly, since path differences (which in turn means differences in heat flow and/or work flow) have no effect on the final state of the system, they instead always manifest themselves as differences in the final state of the surroundings.