For die casting, it is post processing. After the part comes out of the mold there is the sprue, overflow tabs, and flash (extra material at the parting line) that needs to be cleaned up. Then the parts are often chromate conversion coated and then finished with either a wet paint or powder coat. Plating is also an option.
For injection molding, parts typically come out of the mold fully finished (or just requiring gate removal). The vast majority of parts are only textured where the final finish is part of the mold. You can do anything from a high polish to rough textures, or even any crazy pattern or simulated material you can imagine. For graphics you can apply in-mold decoration where a film is placed in the mold before the plastic is injected. There are some post processing finishes that are pretty frequently done such as wet painting and plating.