The identification of universal properties from minimally processed data sets is one goal of machine learning techniques applied to statistical physics. Here, we study how the minimum number of variables needed to accurately describe the important features of a data set - the intrinsic dimension ($I_d$) - behaves in the vicinity of phase transitions. We employ state-of-the-art nearest neighbors-based $I_d$-estimators to compute the $I_d$ of raw Monte Carlo thermal configurations across different phase transitions: first-, second-order and Berezinskii-Kosterlitz-Thouless. For all the considered cases, we find that the $I_d$ uniquely characterizes the transition regime. The finite-size analysis of the $I_d$ allows not just to identify critical points with an accuracy comparable with methods that rely on {\it a priori} identification of order parameters, but also to determine the corresponding (critical) exponent $ν$ in case of continuous transitions. For the case of topological transitions, this analysis overcomes the reported limitations affecting other unsupervised learning methods. Our work reveals how raw data sets display unique signatures of universal behavior in the absence of any dimensional reduction scheme, and suggest direct parallelism between conventional order parameters in real space, and the intrinsic dimension in the data space.