Nothing to to with CAT I or CAT II/III. Non-precision approaches have final approach fixes, precision approaches have final approach points.
The final segment of a precision approach starts where the nominal glide path intersects the intermediate flight altitude. However, the height of this altitude will vary depending on atmospheric conditions. On a cold day, an altitude of 3000 feet will be lower than a geographical height of 3000 feet, higher on a warm day. Hence, the glide path will be intercepted at different points and it doesn't make sense to publish a fix.
In a non-precision approach, on the other hand, there is no vertical guidance, i.e. no glide path. The final approach begins over the final approach fix, which is a defined geographical location and marked on approach charts with the Maltese Cross.