To account for joint tracking and classification (JTC) of multiple targets from observation sets in presence of detection uncertainty, noise and clutter, this paper develops a new trajectory probability hypothesis density (TPHD) filter, which is referred to as the JTC-TPHD filter. The JTC-TPHD filter classifies different targets based on their motion models and each target is assigned with multiple class hypotheses. By using this strategy, we can not only obtain the category information of the targets, but also a more accurate trajectory estimation than the traditional TPHD filter. The JTC-TPHD filter is derived by finding the best Poisson posterior approximation over trajectories on an augmented state space using the Kullback-Leibler divergence (KLD) minimization. The Gaussian mixture is adopted for the implementation, which is referred to as the GM-JTC-TPHD filter. The L-scan approximation is also presented for the GM-JTC-TPHD filter, which possesses lower computational burden. Simulation results show that the GM-JTC-TPHD filter can classify targets correctly and obtain accurate trajectory estimation.