You shouldn't think about the cube in terms of tiles, but on the mini-cubes it is split into, sometimes called cubies or cubelets. There are 26 of these (3*3*3 minus the center), and there are 3 types. The centers (1 color), the edges (2 colors), and the corners (3 colors). For instance, it is not just a blue tile, it is a blue-red edge piece, which there is only one of.
the center pieces can't be moved, so you solve the cube around these. For instance, the the orange-white-green corner has to be placed in the corner between the orange, white, and green centers. It also has to be rotated so the colors match.
To answer your question, then yes, every piece, and therefore every tile on them, has a single unique position on a solved cube. The only thing that can change is the center pieces. While they cannot be moved, they can be rotated, and since they are symmetrical, you can't tell if they have been. For some variations of the cube, the centers have to be oriented too.