Plane stress and plane strain are two different simplifications of reality. In my opinion a problem studied in plane stress is very difficult that become in plain strain, unless the problem changes the assumptions of the formulations.

For example... the statics of a retaining wall can be studied in plan strain (translational symmetry). But if (after a period of time) occur localized settlements foundations (that broke the translational symmetry), this behaviour can probably be studied in plane stress.

One example I can remember is the analyses of a rocket propellant grain. If the propellant grain has a star configuration running throughout the length it can be analysed as a plane stress problem. While if the propellant grain has number of circumferential radial slots along the length that is axisymmetric then it is better to analyse by plane strain assumption. The assumptions are mainly made by understanding the physical problem as to what stress will govern the failure or design criteria. Here the stresses are relevant to the sharp corners and the loading pattern remaining the same for the slice of grain taken. In the plane stress the slice is dx at any length, while for the plane strain it is the longitudinal a I symmetric slice taken for finite element or design analyses.

They are not the same. Plane stress means out-of plan stresses is zero. This is the case of beam subjected to in-plan load. Plane strain means the out-of-plane stresses are zero as in the case of shear wall.