The collapse of multiplicity distributions P n onto a universal scaling curve arises when P n is expressed as a function of the standardized multiplicity ( n− c)/ λ with c and λ being location and scale parameters governed by leading particle effects and the growth of average multiplicity. It is demonstrated that self-similar multiplicative cascade processes such as QCD parton branching naturally lead to a novel type of scaling behavior of P n which manifests itself in Mellin space through a location change controlled by the degree of multifractality and a scale change governed by the depth of the cascade. Applying the new scaling rule it is shown how to restore data collapsing behavior of P n measured in hh collisions at ISR and SPS energies.