This paper derives a rigorously Bayesian technique for estimating the identities of a plurality of targets that are well separated or tracked using the (joint) probabilistic data association filter. In contrast to the single-target classification problem, the joint identification of multiple targets is characterized by statistical dependencies between track-to-identity assignments that render track-level estimation of identity suboptimal. The present method rigorously accounts for these dependencies and allows arbitrary feature and kinematic measurements generated by individual targets to be used in finding the statistically-optimal track-to-identity assignment probabilities. The problem is decomposed into global combinatorial identity deconfliction and local target tracking and classification that is based on a unified measure-theoretic filtering framework. The computational complexity of this technique is shown to be dominated by calculation of the permanent of a non-negative matrix, which may be found exactly in exponential time or approximated in polynomial time using Markov chain Monte Carlo methods. Strategies for improving numerical performance are given for cases where certain subsets of targets are indistinguishable or unobservable. This work is relevant to applications in tactical settings, surveillance, including video tracking, air/land/maritime situational awareness, and automated intelligence collection.