Cluster morphology of spherical particles interacting with a short-range attraction has been extensively studied due to its relevance to many applications, such as the large-scale structure in amorphous materials, phase separation, protein aggregation, and organelle formation in cells. Although it was widely accepted that the range of the attraction solely controls the fractal dimension of clusters, recent experimental results challenged this concept by also showing the importance of the strength of attraction. Using Monte Carlo simulations, we conclusively demonstrate that it is possible to reduce the dependence of the cluster morphology to a single variable, namely, the reduced second virial coefficient, B2, linking the local properties of colloidal systems to the extended law of corresponding states. Furthermore, the cluster size distribution exhibits two well-defined regimes: one identified for small clusters, whose fractal dimension, df, does not depend on the details of the attraction, i.e., small clusters have the same df, and another related to large clusters, whose morphology depends exclusively on B2, i.e., df of large aggregates follows a master curve, which is only a function of B2. This physical scenario is confirmed with the reanalysis of experimental results on colloidal-polymer mixtures.