In the case of fractals the number of dimensions has to do (this is a little hand-wavy) with the way points in the fractal are distributed. You can have a set that’s really d-dimensional, but on a large scale it appears to be a different dimension. That all said, when somebody (looking at you, all mathematicians) talks about fractional dimensions, they’re really talking about a weird, abstract, and not at all physical notion of dimension. When we talk about the “dimension of space” we’re talking about the number of completely different directions that are available, not the whole “sphere thing”. The dimensions of space are either there or not, so while you could have 4 dimensions, you couldn’t have 3.5.