Urban-induced microclimate variations, such as urban heat islands and air pollution, scale with city size, producing distinctive relations between average climate variables and city-scale quantities (e.g., total population). However, these relations are sensitive to city boundary definitions and overlook intra-urban variability. Here, we overcome these limitations by using high-resolution data of urban temperatures, air quality, population, and street networks from 142 cities worldwide, showing that their marginal and joint probability distributions collapse onto a set of general functions inspired by finite-size scaling in statistical physics. Through a logarithmic relation linking urban spatial features to climate variables, we find that average street network properties are sufficient to characterize the full variability of temperature and air pollution fields within and across cities. These findings show that intra-urban climate variability follows general scaling functions, enabling the integration of climate information into reduced-complexity models of urban systems to better inform future urban planning.