I think that's a misleading characterization. It's actually a particularization not a generalization: the ordinary notion only deals with topological properties

I think that's a misleading characterization. It's actually a particularization not a generalization: the ordinary notion only deals with topological properties, fractal/Hausdorff dimension deals with metric properties as well. That's the main distinction: the old notion doesn't care about roughness or changes in measurement at varied scales since it has nothing to do with metric properties. The generalization from integers to reals in the actual values is not indicative of the relationship between the two concepts.