I am reading a physics textbook which implicitly uses this assumption in dealing with photon electron scattering. In general, I don't see why this is true. I can imagine the first two particle momenta spanning the x-y plane and the second two spanning the x-z plane, for instance, as long as the z-components of the final momenta are equal and opposite.
You are right about there being no reason for 2-particle collisions to be restricted to a plane. However, if one of the particles has mass, then there is always a frame of reference where the action happens in a plane. For the electron-photon collision, if you go to the rest frame of the electron, then there are three momentum vectors: the photon's pre-collision momentum, the photon's post-collision momentum, and the electron's post-collision momentum. To conserve momentum, these three vectors have to lie in the same plane.
If you are studying Compton scattering, then the electrons in experiments are bound to atoms, making them effectively motionless in the lab frame compared to the incoming photon. This makes it valid to study the problem in two dimensions.
No, it is not necessarily in a plane. The textbook has simplified it for analysis. Firstly, if it were always in a plane then that would be major news. Secondly, the cross section (which is perpendicular to particle movement) has to have two dimensions for there to be any chance of collision. So there is at least some amount of a third dimension in their paths.