I haven't used fleece for batting or backing, so this is my conjecture only.
If using fleece as a backing, I probably would not bother with batting unless the batting is extremely thin. Fleece provides some loft. Adding a batting will stiffen the drape of the quilt considerably and add a lot of weight with probably little gain in warmth since fleece is polyester and doesn't "breathe" well anyway. The quilt won't "breathe" any better with a middle layer of batting. Fleece adds a nice softness of feel to the user plus a fleece backing will stop a quilt from slipping and sliding around on a bed so I can see value in using fleece as a backing, especially if batting is omitted.
I see no real advantage to using fleece as batting between a cotton top and cotton backing. Cotton "breathes" whereas fleece does not, so you would be adding a non-breathable layer between two breathing layers. Fleece would not add more warmth than a good cotton or wool batting, but fleece is much stretchier in at least one direction much more so than regular battings of any kind. This stretchiness in the middle likely makes it more difficult to quilt all of the layers without puckers and tucks. Unless cost is a big factor, I just don't see how substituting fleece for batting would be a value-added choice.