Attempts to understand the relationship between diversity, productivity and scale have remained limited due to the scheme-dependent nature of the taxonomies describing complex systems. We analyze the diversity of US metropolitan areas in terms of profession diversity and employment to show how this frequency distribution takes a universal scale-invariant form, common to all cities, in the limit of infinite resolution of occupational taxonomies. We show that this limit is obtained under general conditions that follow from the analysis of the variation of the occupational frequency across taxonomies at different resolutions in a way analogous to finite-size scaling in statistical physical systems. We propose a theoretical framework that derives the form and parameters of the limiting distribution of professions based on the appearance, in urban social networks, of new occupations as the result of specialization and coordination of labor. By deriving classification scheme-independent measures of functional diversity and modeling cities as social networks embedded in infrastructural space, these results show how standard economic arguments of division and coordination of labor can be articulated in detail in cities and provide a microscopic basis for explaining increasing returns to population scale observed at the level of entire metropolitan areas.