All the descriptions below apply to solid elements, not shell elements.
Fixed Support means X, Y and Z are all set to 0.
If you have a plane of symmetry, that sets the displacement normal to that plane to 0.
If the normal to a plane of symmetry was the Z axis, and you put both symmetry and a Z=0 displacement constraint on the face, the result would be that the nodes on that face would have a Z=0 constraint. There is no problem specifying the same constraint twice.
If you have a cylinder and you slice it twice to make a quarter cylinder and have symmetry conditions on the two symmetry planes (say with an X and a Y normal), and you apply Z = 0 to the end face, you have fully constrained the quarter cylinder and can pull on the other end with a force or pressure.
This will allow the Z=0 end to slide in the X and Y directions as the Poisson's ratio takes effect and will result in a nice uniform stress at that end. This is a lot different to the full cylinder that has a fixed support. The face cannot move as the Poisson's ratio dictates and stresses are created because of that.
If you have a full cylinder and only use Z=0 instead of a fixed support, the solver cannot find equilibrium because X and Y are free and it doesn't know where to put them. This is resolved in the quarter model using two planes of symmetry.